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NUMERICAL MODELING OF UNSTEADY AND NON-EQUILIBRIUM SEDIMENT TRANSPORT IN
RIVERS
A Thesis Submitted to The Graduate School of Engineering and Sciences of
İzmir Institute of Technology In Partial Fulfillment of the Requirements for the Degree of
MASTER OF SCIENCE
in Civil Engineering
by Aslı BOR
December 2008
İZMİR
We approve the thesis of Aslı BOR Prof. Dr. Gökmen TAYFUR Supervisor Department of Civil Engineering İzmir Institute of Technology Asist. Prof. Dr. Şebnem ELÇİ Department of Civil Engineering İzmir Institute of Technology Prof. Dr. M. Şükrü GÜNEY Department of Civil Engineering Dokuz Eylül University ….. …. …… ….. 04.12.2008 Date
Prof. Dr. Hasan BÖKE
Dean of the Graduate School of Engineering and Science
Prof. Dr. Gökmen TAYFUR Head of the Department of Civil Engineering
ACKNOWLEDGEMENTS
I would like to thank my advisor Prof. Dr. Gökmen TAYFUR for making this
work possible. I am very grateful for his constant guidance, support and encouragement
throughout my masters program. I would like to extend a note of thanks to Asst. Prof.
Dr. Şebnem ELÇİ for taking her valuable time out and agreeing to be on my thesis
committee and also helping me out in times of need.
This study would not be possible without the support of The Scientific and
Technical Research Council of Turkey (TUBİTAK Project number 106M274). I also
would like to special thanks to Prof. Dr. M. Şükrü GÜNEY, who is the principle
investigator of the project and he is the member of thesis committee for his valuable
assistance throughout this project. I would also like to thank Prof. Dr. Turhan ACATAY
and Gökçen BOMBAR for all the assistance they have provided.
Thanks to research assistants at IYTE Department of Civil Engineering who
helped me in anyway along this study. Thanks to my colleagues Anıl ÇALIŞKAN,
Ramazan AYDIN and Sinem ŞİMŞEK for their help. Also special thanks to Nisa
KARTALTEPE for her encouragement and good friendship.
I would like to extend a note of thanks to my sister Gizem BOR, my cousins
Alptuğ TATLI, Altuğ TATLI and İsmail KESKİN for all their continuous support and
endless love. A special thank to Fatih YAVUZ for his assistance and encouragement.
Finally, I would like to give a special thanks to my parents and all other extend family
members for their love and support, which always inspired me through my research
period.
iv
ABSTRACT
NUMERICAL MODELING OF UNSTEADY AND NON-EQUILIBRIUM SEDIMENT TRANSPORT IN RIVERS
Management of soil and water resources is one of the most critical
environmental issues facing many countries. For that reason, dams, artificial channels
and other water structures have been constructed. Management of these structures
encounters fundamental problems: one of these problems is sediment transport.
Theoretical and numerical modeling of sediment transport has been studied by
many researchers. Several empirical formulations of transported suspended load, bed
load and total load have been developed for uniform flow conditions. Equilibrium
sediment transport under unsteady flow conditions has been just recently numerically
studied. The main goal of this study is to develop one dimensional unsteady and
nonequilibrium numerical sediment transport models for alluvial channels.
Within the scope of this study, first mathematical models based on the
kinematic, diffusion and dynamic wave approach are developed under unsteady and
equilibrium flow conditions. The transient bed profiles in alluvial channels are
simulated for several hypothetical cases involving different particle velocity and particle
fall velocity formulations and sediment concentration characteristics. Three bed load
formulations are compared under kinematic and diffusion wave models. Kinematic
wave model was also successfully tested by laboratory flume data. Secondly, a
mathematical model developed based on kinematic wave theory under unsteady and
nonequilibrium conditions. The model satisfactorily simulated transient bed forms
observed in laboratory experiments. Finally, nonuniform sediment transport model was
developed under unsteady and nonequilibrium flow based on diffusion wave approach.
The results implied that the sediment with mean particle diameter and the sediments
with nonuniform particle diameters gave different solutions under unsteady flow
conditions.
v
ÖZET
NEHİRLERDE KARARSIZ VE DENGESİZ SEDİMENT TAŞINIMININ
NÜMERİK MODELLENMESİ
Toprak ve su kaynakları yönetimi birçok ülkenin karşılaştığı en ciddi çevre
sorunlarından biridir. Bu nedenle barajlar, su kanalları ve diğer su yapıları inşa
edilmektedir. Bu yapıların yönetimi, birçok problemle karşı karşıya kalmaktadır. Bu
problemlerin biri de katı madde taşınımıdır.
Teorik ve nümerik katı madde taşınımı birçok araştırmacı tarafından
çalışılmaktadır. Kararlı akım koşulları altında, askıda katı madde, çökmüş katı madde
ve toplam katı madde taşınımı deneysel formüller yardımıyla geliştirilmiştir. Son
yıllarda, kararsız akım şartları altında dengede katı madde taşınımı nümerik
modellenmesi çalışılan konular arasındadır. Bu çalışmanın amacı da nehirlerde 1
boyutlu, kararsız ve dengesiz sediment taşınımının nümerik modellenmesidir.
Bu amaç çerçevesinde, önce kararsız ve dengeli akım koşulları altında
kinematik, difüzyon ve dinamik dalga yaklaşımına göre üç farklı model geliştirilmiştir.
Alluvial nehirlerdeki geçici yatak profilleri, farklı parçacık hızı ve parçacık düşüm hızı
formülleri ve katı madde karakteristiklerini içeren farklı farazi durumlar için
oluşturulmuştur. Kinematik ve difüzyon dalga yaklaşımı altında üç farklı yatak yükü
formülü karşılaştırılmıştır. Ayrıca kinematik dalga modeli laboratuar verileri ile test
edilmiştir ve sonuçlar başarılı olmuştur. Daha sonraki aşamada, kinematik dalga
yaklaşımını kullanarak kararsız akım şartları altında dengesiz model kurulmuştur.
Kurulan model laboratuar verileri ile test edilmiş ve gözlemlenen yatak profilleri, model
ile başarıyla elde edilmiştir. Son olarak, üniform olmayan katı madde karışımı, difüzyon
dalga yaklaşımı ile kararsız ve dengesiz akım şartları altında modellenmiştir. Sonuçlara
göre katı madde ortalama çapı ile kurulan model ve üniform olmayan katı madde
karışımı ile kurulan model, kararsız akım şartlarında farklı sonuçlar vermiştir. Bu
sonuçlar ayrıca laboratuar verileri ile desteklenmelidir.
vi
TABLE OF CONTENTS
LIST OF FIGURES .......................................................................................................... x
LIST OF TABLES.........................................................................................................xiii
CHAPTER 1. INTRODUCTION ..................................................................................... 1
CHAPTER 2. LITERATURE REVIEW.......................................................................... 6
2.1. Physical Studies ..................................................................................... 6
2.2. Mathematical Studies............................................................................. 8
2.2.1. Analytical Studies ........................................................................... 8
2.2.2. Numerical Studies ........................................................................... 9
2.2.2.1. One Dimensional Model Studies.......................................... 11
2.2.2.2. Two Dimensional Model Studies ......................................... 12
2.2.2.3. Three Dimensional Model Studies ....................................... 14
2.3. Measurement Surveys.......................................................................... 14
CHAPTER 3. MECHANICS OF SEDIMENT TRANSPORT...................................... 16
3.1. Physical Properties of Water .............................................................. 16
3.1.1. Specific Weight ............................................................................. 16
3.1.2. Density .......................................................................................... 16
3.1.3. Viscosity........................................................................................ 17
3.2. Physical Properties of Sediment .......................................................... 17
3.2.1. Size................................................................................................ 18
3.2.2. Shape ............................................................................................. 18
3.2.3. Particle Specific Gravity ............................................................... 19
3.2.4. Fall Velocity.................................................................................. 19
3.2.4.1. Dietrich Approach ................................................................ 21
3.2.4.2. Yang Approach..................................................................... 23
3.3. Bulk Properties of Sediment ................................................................ 23
3.3.1. Particle Size Distribution .............................................................. 23
vii
3.3.2. Specific Weight ............................................................................. 24
3.3.3. Porosity ......................................................................................... 24
3.4. Incipient Motion Criteria ..................................................................... 25
3.4.1. Shear Stress Approach .................................................................. 26
3.4.2. Velocity Approach ........................................................................ 27
3.4.3. Meyer – Peter and Müller Criterion .............................................. 28
3.5. Resistance to Flow with Rigid Boundary ............................................ 28
3.5.1. Darcy – Weisbach, Chezy and Manning formulas........................ 29
3.6. Bed Forms............................................................................................ 31
3.7. Mechanism of Sediment Transport ...................................................... 32
3.7.1. Bed Load Transport Formulas ...................................................... 33
3.7.1.1. DuBoys Approach ................................................................ 33
3.7.1.2. Meyer – Peter’s Approach.................................................... 34
3.7.1.3. Schoklitsch Formula............................................................. 35
3.7.1.4. Shields Approach.................................................................. 36
3.7.1.5. Meyer – Peter and Müller’s Approach ................................. 36
3.7.1.6. Regression Approach............................................................ 37
3.7.1.7. Chang, Simons and Richardson’s Approach ........................ 38
3.7.1.8. Parker et al. (1982) Approach ............................................. 39
3.7.1.9. Tayfur and Singh’s Approach ............................................. 40
3.7.2. Suspended Load Transport Formulas............................................ 41
3.7.2.1. The Rouse Equation ............................................................. 41
3.7.2.2. Lane and Kalinske’s Approach ............................................ 43
3.7.2.3. Einstein’s Approach ............................................................. 44
3.7.3. Total Load Transport Formulas .................................................... 45
3.7.3.1. Einstein’s Approach ............................................................. 45
3.7.3.2. Laursen’s Approach.............................................................. 46
3.7.3.3. Bagnold’s Approach............................................................. 47
3.7.3.4. Engelund and Hansen’s Approach ....................................... 48
3.7.3.5. Ackers and White’s Approach.............................................. 49
CHAPTER 4. ONE DIMENSIONAL HYDRODYNAMIC MODEL........................... 51
4.1. de Saint Venant Equations ................................................................... 51
4.1.1. Continuity Equation in Unsteady Flows ....................................... 52
viii
4.1.2. Momentum Equation in Unsteady Flows...................................... 54
4.2. Kinematic Wave Approximation ......................................................... 59
4.3. Diffusion Wave Approximation .......................................................... 60
4.4. Dynamic Wave Approximation ........................................................... 61
CHAPTER 5. ONE DIMENSIONAL SEDIMENT TRANSPORT MODEL ............... 63
5.1. One Dimensional Numerical Model for Sediment Transport
under Unsteady and Equilibrium Conditions ..................................... 63
5.1.1. Kinematic Wave Model of Bed Profiles in Alluvial
Channels under Equilibrium Conditions....................................... 63
5.1.1.1. Numerical Solution of Kinematic Wave Equations ............. 67
5.1.1.2. Model Testing for Hypothetical Cases ................................. 70
5.1.1.2.1. Hypothetical Case I: Effect of Inflow
Concentration ........................................................... 71
5.1.1.2.2. Hypothetical Case II: Effect of Particle
Velocity and Effect of Particle Fall Velocity ........... 73
5.1.1.2.3. Hypothetical Case III: Effect of Maximum
Concentration ........................................................... 79
5.1.2. Diffusion Wave Model of Bed Profiles in Alluvial
Channels under Equilibrium Conditions....................................... 81
5.1.2.1. Numerical Solution of Diffusion Wave Equation ................ 82
5.1.3. Dynamic Model of Bed Profiles in Alluvial Channels
under Equilibrium Conditions ...................................................... 82
5.1.3.1. Numerical Solution of Dynamic Wave Equations ............... 83
5.1.3.2. Model Testing: Comparing the Kinematic, Diffusion
and Dynamic Models for Hypothetical Cases ..................... 84
5.1.3.3. Hypothetical Case I: Comparing Three Bed Load
Formulas under Kinematic and Diffusion Wave
Models .................................................................................. 87
5.1.3.4. Model Testing Using Experimental Data ............................ 88
5.1.3.4.1. Test I ........................................................................ 88
5.1.3.4.2. Test II ....................................................................... 91
5.2. One Dimensional Numerical Model for Sediment
Transport under Unsteady and Nonequilibrium Conditions ............... 95
ix
5.2.1. Governing Equations..................................................................... 95
5.2.1.1. Numerical Solution of Kinematic Wave Equations ........... 100
5.2.1.2. Model Application.............................................................. 102
5.2.1.3. Model Testing Using Experimental Data .......................... 105
5.2.1.4. Model Testing: Comparing the Equilibrium and
Nonequilibrium models for Hypothetical Cases ................ 107
5.3. One Dimensional Numerical Model for Nonuniform Sediment
Transport under Unsteady and Nonequilibrium Conditions ............. 109
5.3.1. Governing Equations................................................................... 109
5.3.1.1. Numerical Solutions of Nonuniform Model....................... 110
5.3.1.2. Model Application.............................................................. 111
CHAPTER 6. SUMMARY AND CONCLUSION ...................................................... 116
6.1. Summary ............................................................................................ 116
6.2. Conclusion ......................................................................................... 116
REFERENCES ............................................................................................................. 118
APPENDICES
APPENDIX A............................................................................................................... 129
x
LIST OF FIGURES
Figure Page
Figure 1.1. Different modes of sediment transport ....................................................... 3
Figure 3.1. Settling of sphere in still water .................................................................. 25
Figure 3.2. Shields diagram for incipient motion ....................................................... 27
Figure 3.3. Bed forms of sand bed channels ............................................................... 32
Figure 3.4. Movement types of sediment particles ...................................................... 33
Figure 4.1. Definition sketch for continuity equation.................................................. 52
Figure 4.2. Definition sketch for momentum equation................................................ 55
Figure 5.1. Definition sketch of two layer system ...................................................... 64
Figure 5.2. Finite – difference grid .............................................................................. 68
Figure 5.3. (a) Inflow hydrograph (b) Inflow concentration........................................ 71
Figure 5.4. Transient bed profile at (a) rising period (b) equilibrium period
(c) recession period (d) post recession period of inflow
hydrograph and concentration ................................................................... 72
Figure 5.5. Transient bed profile under different particle velocities at
(a) rising period (b) equilibrium period (c) recession period
(d) postrecession period of inflow hydrograph and concentration.
(Source: under Rouse 1938, Dietrich 1982, Yang 1996 formula)............ 77
Figure 5.6. Transient bed profiles under different fall velocities at (a) rising
period (b) equilibrium period (c) recession period
(d) postrecession period of inflow hydrograph and concentration.
(Source: under Chien and Wan 1999, Bridge and Dominic 1984,
Kalinske 1947 formulation. ....................................................................... 78
Figure 5.7. Transient bed profile under different maxz values at (a) rising
period (b) equilibrium period (c) recession period
(d) postrecession period of inflow hydrograph and concentration ........... 80
Figure 5.8. Comparison of numerical solution of Diffusion and Kinematic
waves at distance (a) mx 200= (b) mx 500= (c) mx 800= of
the channel................................................................................................. 85
xi
Figure 5.9. Comparison of numerical solution of Dynamic, Diffusion and
Kinematic waves at distance (a) mx 200= (b) mx 500=
(c) mx 800= (assuming clear water ( )0=c ) ............................................ 86
Figure 5.10. (a) Comparison of Tayfur and Singh, Meyer – Peter and
Schoklitsch bed load formulations under Kinematic wave model
at time 160 min. (b) Comparison of Tayfur and Singh, Meyer –
Peter and Schoklitsch bed load formulations under Kinematic
wave model at distance mx 200= of the channel ................................... 87
Figure 5.11. (a) Comparison of Tayfur and Singh, Meyer – Peter and
Schoklitsch bed load formulations under Diffusion wave model at
time 160 min. (b) Comparison of Tayfur and Singh, Meyer –
Peter and Schoklitsch bed load formulations under Diffusion
wave model at distance mx 200= of the channel .................................... 88
Figure 5.12. (a) The input hydrograph a) Rising limb = 90 second (b) Rising
limb = 120 second ..................................................................................... 89
Figure 5.13. Measured and computed water depths at (a) 10.5 m (b) 14 m
(the hydrograph that has 90 second in rising limb) ................................... 90
Figure 5.14. Measured and computed water depths at (a) 10.5 m (b) 14 m
(the hydrograph that has 120 second in rising limb) ................................. 90
Figure 5.15. Simulation of measured bed profile at (a) 30 min (b) 60 min
(c) 90 min................................................................................................... 92
Figure 5.16. Simulation of measured bed profile at (a) 15 min (b) 45 min
(c) 75 min (d) 105 min............................................................................... 94
Figure 5.17. Definition Sketch of two layer system in nonequilibrium
condition ................................................................................................... 96
Figure 5.18. (a) Inflow hydrograph. (b) Inflow concentration..................................... 103
Figure 5.19. Transient bed profile at (a) rising period (b) equilibrium period
(c) recession period (d) post recession period of inflow
hydrograph and concentration ................................................................. 104
Figure 5.20. Simulation of bed profiles along a channel bed at (a) 30 h,
(b) 60 h, (c) 90 h and (d) 120 h of the laboratory experiment ................. 106
xii
Figure 5.21. Simulation of bed profiles in time during the laboratory
experiment at six different locations of the experimental channel.
Location #1 is 10 m away from the upstream end................................... 107
Figure 5.22. (a) Inflow hydrograph. (b) Inflow concentration..................................... 108
Figure 5.23. Comparing the equilibrium and nonequilibrium models......................... 108
Figure 5.24. Multiply – layer model for bed load column........................................... 109
Figure 5.25. (a) Inflow hydrograph. (b) Inflow concentration .................................... 112
Figure 5.26. Transient bed profiles of nonuniform sediment and uniform
sediment model at (a) rising period (b) equilibrium period
(c) recession period (d) post recession period of inflow
hydrograph and concentration in unsteady flow conditions.................... 113
Figure 5.27. Transient bed profiles of nonuniform sediment and uniform
sediment model at (a) rising period (b) equilibrium period
(c) recession period (d) post recession period of inflow
hydrograph and concentration in steady flow conditions ....................... 114
xiii
LIST OF TABLES
Figure Page
Table 3.1. Properties of water ....................................................................................... 18
Table 5.1. Computed RMSE , MAE , MRE ................................................................. 91
Table 5.2. Computed RMSE , MAE , MRE ................................................................. 93
Table 5.3. Computed RMSE , MAE , MRE ............................................................... 107
Table 5.4. Sediment Characteristics............................................................................ 112
1
CHAPTER 1
INTRODUCTION
River management is as old as human civilization. Since ancient times, rivers
have been used for water supply, flood control, irrigation, tourism, navigation, fishing,
waste disposal and power generation by civilizations. Water is the source of life and soil
is the root of existence. The life cannot exist without water and soil. Water and soil
resources are the most fundamental materials on which people rely for existence and
development. Development of society is determined by its capacity to use its resources.
Some of these resources may in time become exhausted and deteriorate (World
Meteorological Organization 2003). Soil and water are limited and irreplaceable
resources. Especially in developing countries, due to the industrial growth and
urbanization quality and quantity of natural water resources have been rapidly decreased
This may lead to water resources come to an end.
Soil and water losses cause the deterioration of ecology and changes in river
morphology have a direct impact on earth’s landscape. By human activities, as
inappropriate land and water resources usage, land desertification occurs and it makes
the farmland useless forever. Sedimentation is the consequence of a complex natural
process involving soil detachment, entrainment, transport and deposition. It is common
in rivers because of the difference between sediment load and the real sediment
transportation capacity of flow. When sediments are deposited in river basins, the water
level rises and it brings ecological problems such as landslides and slope collapses,
debris flow and flow disasters. It also causes economical problems nationwide. On the
other hand, transport of sediment reduces reservoirs life-time and hydrodynamic
potential of dams and can contribute to contamination of drinking water supplies (Bor,
et al. 2007). Reservoirs are limited, precious and non renewable resources. Reducing the
capacity of the reservoir, affects factors of design aims such as water supply, flood
control, irrigation, and power generation. Sediment accumulation has been estimated to
decrease worldwide reservoir storage by 1% per year (Mahmood 1987). On the other
hand, erosion can cause scouring under the river training works, so it brings some safety
problems for river and it affects water supply and navigation along the rivers.
Furthermore, aggregation and degradation affect the stability of a dam.
2
Sediment particles in water, might behave as a carrier for heavy metals which
have affinity to attach to cohesive sediments. They serve as the major pollutant and can
cause disruption of ecosystems. Sediment particles such as nitrogen, organic
compounds, residues, pathogenic bacteria, pesticides and viruses are carried into a
reservoir, deteriorate water quality and cause different illness (World Meteorological
Organization 2003).
Sedimentation and soil erosion are the modern world’s environmental topics.
These subjects have been studied for centuries by engineers. There are different
approaches for solving engineering problems. Sediment deposition deals with water and
sediment particles so, the physical properties of water and sediment particles should be
studied to understand sediment transport mechanism. Sediments are transported as
suspended and bed load as shown in Figure 1.1 depending upon fundamental properties
of water and sediment particle size, density, etc.
In a river system, loose surface can erode from basin by water and be transported
by stream. Sediment particles can be transported in four modes rolling, sliding, saltation
and suspension. While sediment particles are sliding and rolling, particles continue to be
at contact with the bed. Saltation means that jumping motion along the bed in a series of
low trajectories. Rolling and sliding particles move along the bed surface under the
force of the overlying flow of water. It is often unimportant to distinguish saltation from
rolling or sliding because saltation is restricted to only a few grain diameters in height
(Dyer 1986). A saltating grain may only momentarily leave the bed and rise no higher
than a few (<4) grain diameters. These three modes are bed load transport. Sometimes
sediments stay in suspension for an appreciable length of time called suspended load
transport. Suspension of a sediment grain is one of the modes in water systems that
occurs when the magnitude of the vertical component of the turbulent velocity is greater
than the settling speed of the grain. Bagnold (1966) argued that the major distinction in
sediment transport modes is between suspended and unsuspended (bed load) transport.
Bed load sediment grains and aggregates are transported under the combined processes
of saltation, rolling, and sliding, and receive insufficient hydrodynamic impulses to
overcome gravitational settling. Their only significant upward impulse is derived from
successive contacts with the bed (Dyer 1986). When the flow conditions satisfy or
exceed the criteria for incipient motion, sediment particles along an alluvial bed will
start to move (Yang 1996).
3
Figure 1.1. Different modes of sediment transport
(Source: Singh 2005)
It is essential to predict effects of sedimentation and loss of storage capacity in
advance for better operation of the reservoirs. Current research on reservoir
sedimentation prediction is mainly based on numerical modeling of sediment transport
methodologies (Hotchkiss and Parker 1991) and investigation of transport parameters in
the laboratory (Guy, et. al. 1966, Soni 1981a).
Free-surface flows can be classified into various types using criteria of their
classification (Chaudhry 1993). Steady and unsteady flows based on changes with
respect to time. In steady flow regimes, depth and velocity do not vary with time. If
depth and velocity at a point vary with time, the flow regime is classified as unsteady. It
is possible to transform an unsteady flow into a steady flow by having coordinates with
respect a moving reference in some cases. Studying steady flow is easier than unsteady
flows in governing mathematical models although the real world situation is unsteady
flow. Such a transformation is possible only if the wave shape does not change as the
wave propagates.
One of the other classifications based on changes with respect to space. If the
flow velocity at a given instant of time does not change within a given length of
channel, it is uniform flow. It means that the convective acceleration is zero. If the flow
velocity at a time varies with respect to distance, it is non-uniform flow. Steady and
4
unsteady flows are characterized by the variation with respect to time at a given
location, whereas the uniform and nonuniform flows are characterized by the variation
at a given instant of time with respect to distance.
The flow can be classified based on Reynolds number. If the liquid particles
appear to move in definite smooth paths and the flow appears to be as a movement of
thin layers on top each other, it is laminar flow. In natural channels, in laminar flow
Reynolds number is low than 500 ( )500Re < . The flow is characterized by the irregular
movement of particles of the fluid in turbulent flow, Reynolds number is greater than
600 ( )600Re < . If the flow is that 600Re500 << it is called transient flow.
The other classification is based on Froude number. The Froude Number is a
dimensionless parameter measuring of the ratio of the inertia force on an element of
fluid to the weight of the fluid element - the inertial force divided by gravitational force.
If the flow velocity is equal to celerity, it is critical flow ( )1=rF . If the flow velocity is
less than the critical velocity, it is subcritical flow ( )1<rF . If the flow is supercritical
the flow velocity is greater than the critical velocity ( )1>rF .
Hydraulic engineers generally treat channel in one dimension (1D). 1D flow
means that the longitudinal acceleration is significant, whereas transverse and vertical
accelerations are negligible.
Modeling of sediment transport can be assumed in equilibrium or non
equilibrium conditions. If detachment rate and deposition rate are equal, the flow is in
equilibrium condition. In non equilibrium condition, there is difference between
detachment rate and deposition rate. There is no doubt that natural rivers are mostly in
non equilibrium state. Because the real river systems behave as unsteady flow in non
equilibrium state, treating the system with steady flow in equilibrium state is a
simplification.
The main objective of this study is to develop unsteady and non equilibrium one
dimensional numerical model for sediment transport in rivers. For that aim, first of all
three numerical models were developed using the kinematic wave, diffusion wave and
dynamic wave, for describing the bed profile evolution and movement in alluvial
channels under equilibrium conditions. The models were evaluated by simulating bed
profiles for several hypothetical scenarios. The scenarios involve solving the equations
with different formulations of particle velocity, particle fall velocity, sediment flux and
5
different values of maximum bed elevation. Also, the models tested against measured
flume data and the solutions were compared.
This thesis includes six chapters. Chapter 1 aims to present a brief introductory
background to the research subject. Previous relevant physical and mathematical studies
are reviewed in Chapter 2. Sediment transport formulations are summarized in Chapter
3. The one dimensional hydrodynamic model is described by the governing equations in
Chapter 4. In Chapter 5, one dimensional sediment transport equations are governed in
two categories: equilibrium and nonequilibrium. Also three different wave approaches
were discussed: kinematic, diffusion and dynamic waves. The boundary conditions of
the numerical model used in the study, and the testing of the model are described.
Finally, in Chapter 6, the main results and the conclusions of the study are summarized.
6
CHAPTER 2
LITERATURE REVIEW
Sediment related disasters such as debris flow, landslides and slope collapses are
known to occur naturally, causing social and economically problems in the world.
Hence, the human civilizations study sediment transport to reduce the damages of the
disasters and to maximize the benefits of the water resources structures. The studies of
the sediment transport can be classified in two categories. Physical studies are related to
extensive flume and field observations. Mathematical studies are related to develop
theoretical and numerical methods.
2.1. Physical Studies
Physical studies are done by doing experiments in laboratory flumes or by taking
field observations. It is difficult to represent a river by a laboratory flume; so many
assumptions are usually incorporated in laboratory studies. The laboratory studies are
still important for understanding of basic concepts of river flow and sediment transport.
Many investigators have developed empirical methods to represent sediment transport
phenomena using data obtained in laboratory.
Taking real time observations are better to understand the complex real life river
systems. However it is very difficult to take real time data in the field and sometimes it
is even impossible.
Experimental studies have been mostly done with laboratory flume experiments
(Guy, et al. 1966, Langbein and Leopold 1968, Soni 1981a, Wathen and Hoey 1998,
Lisle, et al. 1997, Lisle, et al. 2001). Also laboratory studies are easier than field
measurements and provide control to particular combinations of initial and boundary
conditions (Curran and Wilcock 2005).
Newton (1951) studied, with a series experiments, degradation with uniform
sediment size. He saw that the bed elevation and bed slope decreased asymptotically
with time.
Leopold and Maddock (1953) obtained field data showing the relationship
between total sediment discharge and water discharge.
7
Lane and Borland (1954) conducted experiments to study riverbed scour during
floods. Laboratory data were obtained for degradation in alluvial channels by
Suryanarayana (1969).
Colby and Hembree (1955) compared the results of total sediment discharge and
water discharge between computed and measured from the Niobrara River near Cody,
Nebraska. Yang (1973) unit stream power equation gave the best agreement with those
measurements.
Bhamidipaty et al. (1971) studied with Newton’s analysis and combined their
own extensive laboratory flume studies for three different particle sizes with uniform
sediment grain sizes. They observed that the bed elevation in a degrading channel
decreases exponentially with time. Soni et al. (1980) conducted a similar experiment
using mobile bed under equilibrium conditions before the aggradation started. Hence
they formed experiment conditions to better present the real river systems and
developed a mathematical model for aggradation in an infinitely long channel. In 1980
Mehta (1980) improved studies by Soni et al. (1980) research with different sediment
size particles.
Vanoni (1971) compared the computed sediment discharges from different
equations with the measured results from natural rivers. Yang and Stall (1976) and
Yang (1977) reported his comparisons.
For aggradation and deggradation of non uniform sediments, Little and Mayer
(1972) conducted a series of experiments. They studied the variations of sediment
gradation on the bed surface during the armoring.
Ribberink (1985) studied the vertical sorting phenomenon of sediment having an
idealized gradation under the equilibrium conditions. He also proposed a transport layer
concept.
Yen et al. (1985) and Yen et al. (1988a) conducted a series of overloading
experiments using uniform coarse sediment and found that the mean sediment transport
velocity and aggradation wave speed increase with the initial equilibrium bed slope and
decrease with loading ration.
Wilcock and Southard (1989) did careful measurements and observations in
equilibrium conditions and investigated the interaction between the transport, bed
surface texture and bed configuration. Also, Wilcock et al. (2001) conducted five
different sediments in a laboratory flume by carrying out 48 sets of experiments of flow,
transport and bed grain size.
8
Yen et al. (1992) also did flume studies with constant median sediment particle
diameter but varying geometric standard deviation, so that the effect of non uniformity
in rivers could be taken into account. They investigated that aggradation and
degradation depends on materials vary so that the effect of non uniformity in rivers
could be taken into account.
Tang and Knight (2006) investigated the effect of flood plain roughness on bed
form geometry, bed load transport and dune migration rate.
Experimental flume studies have the limitations due to the complexity of
representing a real life river conditions. However, it helps us to understand basic
concepts of river flow and it provides a detailed analysis for parameters related to
physics of the problem.
2.2. Mathematical Studies
Both experimental flume studies and field observations have limitations in
predicting sediment transport capacity. Laboratory studies do not represent real life
river conditions, besides taking the survey data sometimes impossible. Due to these
restrictions, investigators have made many assumptions during the research. To study
the sediment transport mechanism, many investigators developed mathematical
equations for real life situations. All the sediment transport mathematical models
developed so far are based on five basic physical equations. These equations have been
developed by many researches that can be solved both analytically and numerically.
Solving the complex differential equations, numerical solutions are more
appropriate than analytical solutions. On the other hand analytical solutions can be
developed and applied only in very simplified and simple cases.
2.2.1. Analytical Studies
When flow conditions are very simplified in one dimensional case analytical
solution can be developed. Developing the solution is very complex for generalized two
or three dimensional cases with complex conditions. Some of the well known analytical
sediment models are summarized below.
9
Tinney (1955) solved one dimensional differential equation analytically to
simulate the degradation of bed composed of uniform sediment in open channel and
compared his result with Newton (1951).
Al-Khalif (1965) developed a bed load function which explains the aggradation
using Einstein (1950) approach.
de Vries (1971) and de Vries (1973) developed a linear hyperbolic bed elevation
change model using convection – acceleration and depth gradients.
Soni et al. (1980) used a linear diffusion model to predict the transient bed
profiles due to sediment overloading.
Jain (1985) studied the process with appropriate boundary conditions.
Begin et al. (1981) computed longitudinal profiles that produced by base- level
lowering using diffusion model.
Jaramillo (1983) estimate bed load discharge for a finite and semi finite domain
using linear parabolic sediment transport model. The bed elevation was estimated using
sediment transport equation. Jaramillo and Jain (1984) developed a nonlinear parabolic
sediment model for non uniform flow and solved the model by using the method of
weighted residuals (Jain 1985). The results were compared with experimental data
obtained by Newton (1951).
Gill (1983a) and Gill (1983b) used Fourier series by the error function methods
and a linear parabolic bed elevation for a finite length channel to solve the linear
diffusion equation for aggradation and degradation.
Zhang and Kahawita (1987) and Gill (1987) solved a nonlinear parabolic
aggradation and degradation model and compared the solutions with experimental and
linear solutions. They presented that the bed elevation is a function of square root of
time.
Mosconi (1988) developed a linear hyperbolic analytical model for aggradation
in the case of increase of sediment discharge and nonlinear parabolic analytical model
for degradation in the case of reduction of sediment discharge.
2.2.2. Numerical Studies
The linear and non linear parabolic equations are generally based on the
assumption of steady state or quasi state water flow. Unsteady water flow makes the
10
system complex and analytical solution is difficult to develop for the complex systems.
Numerical sediment transport models have been developed in one, two or three
dimensional have been listed below.
Lyn and Altinakar (2002) predicted bed elevation using quasi – steady model.
Curran and Wilcock (2005) studied constant flow rate and flow depth while
varying the sand supply.
Mathematical sediment transport models have been based on generally diffusion
wave and dynamic wave to predict bed profiles in alluvial channels. Whereas many
researchers (de Vries 1973, Soni 1981b, Soni 1981c, Ribberink and Van Der Sande
1985, Lisle, et al. 2001) studied diffusion equations, others (Ching and Cheng 1964,
Vreugdenhil and de Vries 1973, de Vries 1975, Ribberink and Van Der Sande 1985,
Pianese 1994, Lyn and Altinakar 2002, Cao and Carling 2003, Singh, et al. 2004,
Mohammadian, et al. 2004, Li and Millar 2007) studied dynamic equations. The
sediment transport function has been expressed as a function of water flow variables
and the bed formation and the bed movement has been treated as having diffusion
characteristics in literature (Tayfur and Singh 2006). On the other hand the experimental
studies by Langbein and Leopold (1968) provided that movement of bed profiles
behaves as kinematic wave, a function of sediment transport rate and concentration.
Kinematic wave theory applicatibility to unsteady flow routing problems is discussed by
Tsai (2003). Tayfur and Singh (2006) used the kinematic wave theory under equilibrium
conditions and modeled transient bed profiles.
Other mathematical approaches are equilibrium and nonequilibrium sediment
transport models. In equilibrium models, the actual sediment transport rate is assumed
to be equal to the sediment transport capacity at every cross section whereas in many
cases the inflow sediment discharge imposed at the inlet is different than the transport
capacity which might lead to difficulties in the calculation of bed changes near the inlet,
thus solved by non- equilibrium models. Calculation of the equilibrium models are
easier than non- equilibrium models. In many studies it was assumed that detachment
rate and deposition rate are equal. This assumption may be valid only if conditions such
as channel geometry, water and sediment properties are constant for a long period of
time. Natural rivers are mostly in non – equilibrium state. Wu et al. (2004) developed
one dimensional numerical model in unsteady flows under non – equilibrium
conditions. Tayfur and Singh (2007) developed a mathematical model using kinematic
wave theory under non – equilibrium conditions in alluvial rivers.
11
2.2.2.1. One Dimensional Model Studies
In rivers, the accelerations in lateral and vertical directions are mostly assumed
negligible and therefore, acceleration in longitudinal direction is generally utilized in
one dimensional models. This assumption simplifies the solution as it involves few
equations only in one direction. These models have been mostly solved based on finite
difference method to obtain bed elevation and water surface profiles (Perdreau and
Cunge 1971, Cunge and Perdreau 1973, Chang 1982, Krishnappan 1985, Rahuel, et al.
1989, Holly and Rahuel 1990a, Holly and Rahuel 1990b, Correia, et al. 1992, Holly, et
al. 1993).
de Vries (1965) has developed one dimensional model using explicit finite
difference scheme to compute bed and water elevation profiles.
Cunge et al. (1980) has developed one dimensional model simulations of alluvial
hydraulics.
Rahuel et al. (1989) studied unsteady flow models and have applied in river
conditions. Cui et al. (1996), Kassem and Chaudhry (1998), Cao and Egiashira (1999),
Capart (2000), Cao et al. (2001), Capart and Young (2002) and Di Cristo et al. (2002)
have studied similar models in resent years, wary numerical models.
The majority of one dimensional unsteady models can be divided into two
categories in the literature: (1)uncoupled flow models that water flow equations and
sediment continuity equation are solved separately and (2)quasi-steady flow models that
energy equation solved with sediment continuity equation. Only a few models are
coupled in literature. Lyn and Goodwin (1987) presented an approach to model fully
coupled unsteady water flow equations and sediment continuity equation. They
compared the solutions between stability of coupled and uncoupled models and
concluded that the coupled model is more stable. Other one dimensional coupled
sediment transport models presented by Rahuel and Holly (1989), Holly and Rahuel
(1990a, 1990b) simulating process between bed load and suspended load. Correria et al.
(1992), studied with full explicit coupling models using water continuity equation, so it
gives the permission to change the bed roughness depending on flow regime.
Bhallamudi and Chaudhry (1989) have presented one dimensional, unsteady and
coupled deformable bed model using Mac Cormack second order accurate explicit
scheme. They compared the results with experimental laboratory flumes data and saw
12
that the results are satisfactory. Singh et al. (2004) have developed a fully coupled one
dimensional alluvial river model and governed system of partial differential equations
using Preissmann finite difference scheme. The tests presented by simulating the Quail
Creek failure in Washington, USA. Wu et al. (2005) proposed one dimensional model
simulates under unsteady flow conditions in dendritic channel networks with hydraulic
structures. The equations solved in a coupling model and tested in several cases.
Although in uncoupled models, there is strong interaction between solid and
water phases of the flow, only the flow continuity and momentum equations are solved
simultaneously (Singh, et al. 2004). Park and Jain (1986) used Preissmann linearized
implicit scheme for simulating the governing equations in unsteady and uncoupled
models. Lyn (1987) studied uncoupled models and suggested that complete coupling
between the full unsteady flow equations and sediment continuity equation is desirable
in cases where the conditions change rapidly at the boundaries.
2.2.2.2. Two Dimensional Model Studies
Sediment concentration is averaged only along one direction, generally vertical
direction (depth – integrated) where vertical variations are not significant depending on
the flow characteristics in two dimensional models. One of the advantages of the 2D
simulation of flow and sediment transport is depth – averaged subsystem for river flow.
In depth – averaged models, all the model parameters are assumed to be same
everywhere the water column. The depth - integrated equations of motion and
continuity are linked to a depth - integrated sediment transport model (Boer, et al. 1984,
McAnally, et al. 1991). The two dimensional models are more difficult than the one
dimensional models and they provide more information about flow conditions.
Although the best mathematical model is the three dimensional it is not practical
since it requires much more computational time especially in longer river stretches. In
addition, enough experimental data cannot be available in general for model calibration.
Struiksma (1985) and Shimizu and Itakura (1989) developed a two dimensional
model for the simulation of the large scale bed change in alluvial channels.
Mohapatra and Bhallamudi (1994) developed two dimensional model using a
false transient principle with the quasi – steady uncoupled approach in a transformed
coordinate system and McCormack scheme was used for the numerical solution.
13
Chaudhary (1996) developed the model for straight and meandering channels.
MIKE21 (DHI 2003), TABS-MD (Thomas and McAnally 1990), CCHE2D (Wu
2001) and HSCTM2D (Hayter 1995) are the widely used two dimensional sediment
transport models.
MIKE21C is the curvilinear finite difference model. It has been developed at the
Danish Hydraulic Institute (DHI) for river morphology (Langendoen 1996). The effects
of secondary flow are taken into account by introducing quasi – steady approach in
curved channels. In bends, the direction of sediment transport has been determined by
using this secondary flow. Also, the model has been used to simulate critical
morphological and hydrodynamic conditions.
One of the depth – integrated two dimensional sediment transport model is
CCHE2D (Wu 2001). This model is based on variantion of the finite element method
using depth – average ε−k models to estimate the turbulent eddy viscosity. The
secondary flow effects were modeled on bed load direction in curved channels although
fluid momentum and sediment transport rate effects were not. This model is applicable
for morphological problems in rivers.
HSCTM2D (Hydrodynamic, Sediment and Contaminant Transport Model)
model was developed for U. S. Environmental Protection Agency which based on the
finite element method and vertically integrated in cohesive sediments.
Other well known models for simulation of sediment transport are TRIM-2D
(Casulli 1990) based on finite difference approach and was adapted for practical
applications. MOBED2 (Spasojevic and Holly 1990) models with finite difference and
applicable in natural rivers, and TELEMAC2D with its module TSEF based on standard
equilibrium bed load formulations as Meyer – Peter Müller (1948) uses ε−k models
with finite element model.
Minh Duc et al. (2004) developed a depth – averaged model using a finite
volume method to calculate bed deformation in alluvial channels.
Li and Millar (2007) studied two dimensional hydrodynamic bed model to
simulate bed load transport.
14
2.2.2.3. Three Dimensional Model Studies
The sediment transport process in alluvial channels could be described best by
three dimensional models that include all the space dimensions. Since the full equations
of motion are solved, the model is the most complicated and resource consuming in
implementation. When the flow is stratified in salinity or temperature, mostly three
dimensional models are applicable.
Demuren and Rodi (1986) used ε−k models to develop neutral tracer transport
model.
Wan and Adeff (1986) developed finite element method for unsteady flow.
Van Rijn (1987) developed equations for mass balance using three dimensional
equations and combined them with two dimensional depth integrated model.
Lin and Falconer (1996) developed a three dimensional sediment transport
model for estuaries and coasts.
Hamrick (1996) developed EFDC and tested numerical model. EFDC can
simulate flow processes in all three dimensions in rivers, lakes, reservoirs, estuaries,
wetlands and coastal regions.
Wu (2000) studied three dimensional models for straight channels.
Delft-3D (Delft Hydraulics 2002) and ECOMSED (HydroQual Inc. 2003) are
general three dimensional models that are used widely.
ECOMSED is the sediment transport model that was developed by HydroQual,
Inc. and Delft Hydraulics (Blumberg and Mellor 1987) for estuaries and oceans. This
model is applicable only up to a diameter size of 500 mμ and cannot be applied for bed
load transport. HydroQual, Inc. developed the SED module of ECOMSED (2002). A
three dimensional suspended sediment transport model is SED formed for non cohesive
sediments using implicit scheme.
2.3. Measurement Surveys
The geomorphologic data of river can be obtained by a topographic survey,
including a land survey and groundwater surveying, or by repetitive surveying with pre-
determined ranges, since samples size distribution can be found and determined the dry
density or unit weight. Also, surveying of reservoirs are required to determine
15
sedimentation rates and to assess overall capacity of the reservoir. For surveying,
manual sounding poles, sounding weights and echo sounders are commonly used. For
reasons of economy, accuracy and expediency, sedimentation surveys were carried out
in small reservoirs or cross small river reaches. More advanced instruments have been
adopted as electronic distance measuring systems for large reservoirs. Sedimentation
surveys are best reliable for the accurate positioning of measuring points where no
deposition or erosion takes place, the elevation of the bed surface should coincide with
that measured in a previous survey (Bor, et al. 2008). This is a good check of the
accuracy and reliability of the sedimentation surveys. In addition to this detailed
bathymetry map, thickness and long-term average accumulation rates of the lake can be
determined by using echo sounder systems (Odhiambo and Boss 2004). Other studies in
literature about surveying using acoustic methods include the technical details of
scanning (Urick 1983), techniques used for sediment mapping (Higginbottom, et al.
1994), and the comparison of different echoes on sediment type (Collins and Gregory
1996).
Also, in hydrometric stations for sediment measurement, suspended sediment
discharge and sediment concentration, size gradation of suspended sediment and bed
material can be measured the whole year around.
Taking real time observations can explain the real life systems better than flume
experiments but it is very difficult to take real time data in the field even sometimes it is
impossible.
16
CHAPTER 3
MECHANICS OF SEDIMENT TRANSPORT
Sediment transport mechanism is concerned about water and sediment particles.
An understanding of the sediment transport mechanism requires the learning of the
physical properties of water and sediment particles. Fundamental properties of water
and sediment particles are described below.
3.1. Physical Properties of Water
The fundamental properties of water are important in sediment transport studies.
They are summarized below.
3.1.1. Specific Weight
Specific weight is defined as weight per unit volume. Specific weight can be
expressed as (Yang 1996):
gργ = (3.1)
where,
γ =specific weight (M/L2/T2)
ρ =density (M/L3)
g =gravitational acceleration (L/T2)
3.1.2. Density
Quantity of matter contained in a unit volume of the substance.
=ρ vm / (3.2)
17
where,
m =mass (M)
v =volume (L3)
3.1.3. Viscosity
Due to cohesion and interaction between molecules, resistance to deformation is
observed. Viscosity of the property defines the rate of this resistance to deformation.
Newton’s law of viscosity relates shear stress and velocity gradient by dynamic
viscosity.
dyduμτ = (3.3)
where,
τ =shear stress (M/L2)
μ =dynamic viscosity (M / (LT))
dydu
=velocity gradient
Kinematic viscosity is the ratio between dynamic viscosity and fluid density
(Yang 1996).
ρμυ = (3.4)
where,
υ =kinematic viscosity (L2/T)
The properties of water are summarized in Table 3.1.
3.2. Physical Properties of Sediment
Particle size, shape specific gravity and fall velocity are important for
understanding of sediment transport mechanism.
18
3.2.1. Size
Particle size clearly describes the physical properties of the sediment particle, so
it is the most important parameter for many practical purposes. The sediment size can
be measured by various methods such as sieve analysis, optical methods or visual
accumulation tube analysis. The sediment grade scale suggested by Lane (1947), as
shown in Table 3.1. It was adopted by American Geophysical Union and is still used by
hydraulic engineers.
Table 3.1. Properties of water
(Source: Yang 1996)
Temperature (0C)
Specific Weight
γ (kN/m3)
Density ρ (kg/m3)
Dynamic Viscosity
μ x 103 (N-s/m2)
Kinematic Viscosity
v x 10-6 (m2/s)
0 9.805 999.8 1.781 1.785 5 9.807 1000.0 1.518 1.519 10 9.804 999.7 1.307 1.306 15 9.798 999.1 1.139 1.139 20 9.789 998.2 1.002 1.003 25 9.777 997.0 0.890 0.893 30 9.764 995.7 0.798 0.800 40 9.730 992.2 0.653 0.658 50 9.689 988.0 0.547 0.553 60 9.642 983.2 0.466 0.474 70 9.589 977.8 0.404 0.413 80 9.530 971.8 0.354 0.364 90 9.466 965.3 0.315 0.326 100 9.399 958.4 0.282 0.294
3.2.2. Shape
Particle shape is the second most significant sediment property in natural
sediments. The geometric configuration defines shape parameter regardless of sediment
particle size and composition. Grains are usually considered to have with long diameter
19
a, intermediate diameter b and short diameter c. Corey (Schulz, et al. 1954) investigated
several shape factors and defined the shape factor as:
abcCSF = (3.5)
Corel shape factor was the most significant expression of shape. The shape
factor for a sphere would be 1.0. Natural sediment typically has a shape factor of about
0.7 (US Army Corps of Engineers 2008).
3.2.3. Particle Specific Gravity
Specific gravity is defined as the ratio of the specific weight of the sediment to
that of water. It usually ranges numerically from 2.6 to 2.8 in natural solids. While the
lower values of specific gravity are typical of the coarser soils, higher values are typical
of the fine – grained soil types. Due to its resistance to weathering and abrasion, quartz,
which has a specific gravity of 2.65, is the most common mineral found in natural
noncohesive sediments. Typically, the average specific gravity of a sediment mixture is
close to that of quartz. Therefore, in sedimentation studies, specific gravity is frequently
assumed to be 2.65, although whenever possible, site-specific particle specific gravity
should be determined (US Army Corps of Engineers 2008).
3.2.4. Fall Velocity
Fall velocity or settling velocity is the most fundamental property governing the
motion of the sediment particle in a fluid. It is a function of the volume, shape and
density of the particle and the viscosity and density of the fluid. The fall velocity of any
naturally worn sediment particle may be calculated if the characteristics of the particle
and fluid are known. Fall velocity is related to relative flow conditions between the
sediment particle and water during conditions of sediment entrainment, transportation
and deposition. Fall velocity can be calculated from a balance between the particle
buoyant weight or submerged weight and the resulting force from fluid drag (Yang
1996). The general drag equation is
20
2
2f
DD
vACF ρ= (3.6)
where,
DF = drag force
DC = drag coefficient
ρ = density of water
A = the projected area of particle in the direction of fall
fv = the fall velocity
The particle buoyant weight or submerged weight of a spherical sediment
particle is
grW ss )(34 3 ρρπ −= (3.7)
where,
sW =submerged weight
r =particle radius
sρ and ρ = densities of sediment and water respectively.
For very slow, steady moving sphere, the drag coefficient thus obtained is
Re24
=DC (3.8)
This equation is acceptable for Reynolds numbers less than 1.0 where;
υ
sf dv=Re (3.9)
where,
υ = kinematic viscosity of water
sd =sediment diameter
From Equation 3.6 and Equation 3.8, Stokes (1851) equation can be obtained;
21
υρπ fD vdF 3= (3.10)
Equality of Equation 3.7 and Equation 3.10, the fall velocity for a sediment
particle can be obtained as below:
υγ
γγ 2
181 s
w
wsf
dgv −= (3.11)
where,
sγ and wγ = specific weights of sediment and water respectively.
This equation is acceptable for the particle diameter equal to or less than 0.1
mm.
The drag coefficient of a sphere depends on the Reynolds number. When the
particle Reynolds number is greater than 2.0, the particle fall velocity is determined
experimentally. Rouse (1937) gave smv f /024.0= for most natural sands, the shape
factor is 0.7 and mmds 2.0= .
There are many approaches about fall velocity in literature. Some of them
summarized below:
3.2.4.1. Dietrich Approach
Dietrich (1982) developed the following equation for fall velocity analyzing
empirical relation.
)(3*
2110 RRRW += (3.12)
where,
*W = the dimensionless fall velocity
υρρρ
gv
Ws
f
)(
3
* −= (3.13)
22
4
*
3*
2**1
)(log00056.0
)(log00575.0)(log0982.0)(log929.1767.3
D
DDDR
+
−−+−= (3.14)
)6.4(log)1)(5.0(3.0
]6.4tanh[log)1(85.0
)1(1log
*2
*3.2
2
−−−+
−−−⎥⎦⎤
⎢⎣⎡ −−=
DCSFCSF
DCSFCSFR (3.15)
⎥⎦⎤
⎢⎣⎡ −+
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−=
5.2)5.3(1
*3 ]6.4tanh[log83.2
65.0
P
DCSFR (3.16)
where,
*D = the dimensionless particle size
CSF = the Corey shape factor
The dimensionless particle size *D is expressed as (Dietrich 1982):
2
3
*)(
ρυρρ ss gd
D−
= (3.17)
The Corey shape factor (CSF) is expressed as (Dietrich 1982):
5.0)(abcCSF = (3.18)
where,
a,b,c= the longest intermediate and shortest axes of the particle respectively and
mutually perpendicular.
* 8.05.0 −≈CSF (Dietrich 1982)
P = Powers value of roughness ( P is between 3.5 and 6 (Dietrich 1982))
23
3.2.4.2. Yang Approach
Yang (1996) expressed the fall velocity of particle (for the particle Reynolds
number ( )υsfpn dvR = is less than 2.0):
( )
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎥⎦
⎤⎢⎣
⎡ −
−
=
s
w
wss
s
w
ws
f
d
gdF
gd
v
32.3
181
5.0
2
γγγ
υγγγ
for
mmd
mmdmm
mmd
s
s
s
0.2
0.21.0
1.0
>
≤<
≤
(3.19)
where,
( ) ( )⎪⎩
⎪⎨
⎧⎥⎦
⎤⎢⎣
⎡−
−⎥⎦
⎤⎢⎣
⎡−
+=
79.0
363632
5.0
3
25.0
3
2
wss
w
wss
w
gdgdF γγγυ
γγγυ
for mmdmm
mmdmm
s
s
0.20.1
0.11.0
≤<
≤< (3.20)
3.3. Bulk Properties of Sediment
Three important bulk properties are described below, particle size distribution,
specific weight and porosity.
3.3.1. Particle Size Distribution
Particle sizes are determined using a variety of methods. Diameters of particles
larger than 256 mm may be obtained by measuring the mean diameter. Templates with
square openings can be used to determine a size equivalent to the sieve diameter for
particles between 32 and 256 mm. Sieve analyses are generally used for particles
between 0.0625 and 32 mm. For sediments less than 0.0625 mm hydrometer analysis
can be utilized.
24
The variation in particle sizes in a sediment mixture is described with a
gradation curve, which is a cumulative size-frequency distribution curve showing
particle size versus accumulated percent finer, by weight. It is common to refer to
particle sizes according to their position on the gradation curve. d50 is the geometric
mean particle size; that is, 50 percent of the sample is finer, by weight; d84.1 is 1
standard deviation larger than the geometric mean size in practice and it is rounded to
d84, while d15.9 is 1 standard deviation smaller then the geometric mean size and it is
rounded to d16 in practice (US Army Corps of Engineers 2008).
3.3.2. Specific Weight
Specific weight of deposited sediment is the weight per unit volume. It is
expressed as dry weight.
wd SGp γγ )1( −= (3.21)
or
sd p γγ )1( −= (3.22)
where,
dγ = specific weight of deposited sediment
SG = specific gravity of sediment particle
p =porosity
Specific weight increases with time after initial deposition. It also depends on
the composition of the sediment mixture (US Army Corps of Engineers 2008).
3.3.3. Porosity
It is defined as the ratio of volume of voids to total volume of sample. Porosity is
affected by particle size, shape and degree of compaction.
25
t
v
VV
p = (3.23)
where,
vV =void volume
tV =total volume of sample
3.4. Incipient Motion Criteria
The concept of incipient motion of sediment particles from the bed is important
to understand the aggradation and degradation forces acting on a spherical sediment
particle shown in Figure 3.1. The component of gravitational force in the direction of
flow can be neglected compared to other forces acting on a spherical sediment particle
because the channel slopes are small enough in most natural rivers. Other forces are
drag force DF , lift force LF , submerged weight SW and resistance force RF . A sediment
particle starts the incipient motion when the conditions are satisfied below (Yang 1996).
Figure 3.1. Settling of sphere in still water
(Source: Yang 1996)
where,
RO
RD
SL
MMFFWF
===
26
OM =overturning moment due to DF and RF
RM =resisting moment due to LF and SW
Different researchers developed several approaches defining the incipient
motion of sediment particles.
3.4.1. Shear Stress Approach
In early 1936, Shields (1936) derived a function for incipient motion of sediment
particles where balance of forces acting on a particle on a bed was considered. He
applied dimensional analysis to determine dimensionless parameters and investigated
the relationship between these two parameters by experimental studies.
⎟⎠⎞
⎜⎝⎛=
− υγγτ s
sws
c dUfd
*
)( (3.24)
υsdU*Re = (3.25)
where,
Re =Reynolds number
*U = shear velocity
cτ = critical shear stress at initial motion
Vanoni (1975) developed diagram fitting the curve to the data provided by
Shields (Figure 3.2). Figure 3.2. shows the results of experiments about relationship
between dimensional shear stress and particle Reynolds number.
Shields simplified the problem by neglecting the lift force and considered only
the drag force. The American Society of Civil Engineers Task Committee on the
Preparation of Sediment Manual modified diagram and uses a third parameter as shown
in Figure 3.2. The parameter is
2/1
11.0⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛− s
w
ss gddγγ
υ (3.26)
27
This parameter allows determination of intersection with the Shields diagram
and its corresponding values of shear stress. Many investigators have proposed different
options which are more or less the same.
Figure 3.2. Shields diagram for incipient motion
(Source: Vanoni 1975)
3.4.2. Velocity Approach
Velocity approach uses the relationship between the velocity field and shear
stress field. It means that the velocity for incipient motion can be calculated if the drag
force for incipient motion is known.
Yang (1973) obtained incipient motion criteria using laboratory data collected
by different investigators. The relationship between dimensionless critical average flow
velocity and Reynolds number follows a hyperbola when the Reynolds number is less
than 70 summarized in Equation 3.27. When the Reynolds number greater than 70,
ω/crV becomes a constant, as shown Equation 3.28. (Singh 2005).
( ) ,06.006.0/log
5.2
*
+−
=vdUv
V
sf
cr 702.1 * <<υ
sdU (3.27)
,05.2=f
cr
vV
vdU s*70 ≤ (3.28)
28
3.4.3. Meyer – Peter and Müller Criterion
Meyer – Peter and Müller (1948) obtained bed load equation and sediment size
at incipient motion as formulated from bed load equation (Yang 1996).
( ) 2/36/1901 / dnK
SDds = (3.29)
where,
sd =sediment size in the armor layer (in mm)
S =channel slope
D =mean flow depth
1K =constant (=0.9 when D is in ft and 0.058 when D is in m)
n =channel bottom roughness or Manning’s roughness coefficient
90d =bed material size where 90% of the material is finer (in m)
3.5. Resistance to Flow with Rigid Boundary
Prandtl’s (1926) mixing theory depends on velocity distribution approach.
Velocity at distance y is
*log75.55.8 Ukyus⎟⎟⎠
⎞⎜⎜⎝
⎛+= (3.30)
and
**log75.55.5 UyUu ⎟⎠⎞
⎜⎝⎛ +=
υ (3.31)
where,
u =velocity at a distance y above the bed
gDSU =* =shear velocity
29
D =depth of flow
S =slope
sk =equivalent roughness defined by Schlichting (1955)
For sand bed channels,
65dks = (Einstein 1950),
90dks = (Meyer – Peter and Müller 1948),
85dks = (Simons and Richardson 1966, Yang 1996).
3.5.1. Darcy – Weisbach, Chezy and Manning formulas
The Darcy – Weisbach formula for pipe flow is
gV
DLfh f 2
2
= (3.32)
For open channel flow,
42/
2/4
2
DD
D
PA
R ===π
π
(3.33)
and
Lh
S f= (3.34)
So we can express the f value;
2
8VgRSf = (3.35)
2*UgRS = (3.36)
30
2/1
*
8⎟⎟⎠
⎞⎜⎜⎝
⎛=
fUV (3.37)
where,
fh =friction loss
f =Darcy – Weisbach friction factor
L =pipe length
D =pipe diameter
V =average flow velocity
R =hydraulic radius
S =energy slope
The Chezy formula is
RSCV z= (3.38)
Shear stress along the boundary is
20 8
1Vfρτ = (3.39)
From relationship between RU ,, *0τ and V , Chezy coefficient can be obtained by
2/1
8⎟⎟⎠
⎞⎜⎜⎝
⎛=
ργ
fCz (3.40)
The Manning formula is
2/13/21SR
nV = (3.41)
where,
n =Manning coefficient and can be obtained by the formula below;
31
1.21
6/1dn = (3.42)
where,
d =sediment diameter of uniform sand in m (Yang 1996).
3.6. Bed Forms
Rate of sediment transport mainly depends on resistance to flow and bed
configuration. Simons and Richardson (1960) summarized bed forms as shown in
Figure 3.3. below (Yang 1996).
Ripples begin to form, as current velocity picks up in lower flow regime. These
are small bed forms, generally wavelengths less than 30 cm and heights less than 5 cm.
In faster currents, ripples grow into dunes. Dunes are similar to bars but larger than
ripples. Their profile height is limited by depth of flow, so they can be several meters
tall in deep water. Bars are bed forms having lengths the same as channel width and
height same as channel height. In higher velocities dunes are destroyed and plane bed
forms occur. In very high velocities anti dune bed forms occurs. Water surface forms
waves that move upstream and so anti dunes move upstream. These are also called
standing waves. In large slopes, high velocities and sediment concentrations chutes and
pools occur. They consist of large elongated mounds of sediment. In transition zone,
bed configurations range from dunes to plane beds or to anti dunes.
32
Figure 3.3. Bed forms of sand bed channels
(Source: Simon and Richardson 1966)
3.7. Mechanism of Sediment Transport
Sediments are eroded from basin by water and transported by stream when the
flow conditions exceed the criteria for incipient motion. The motion can be rolling,
sliding or jumping along the bed which is called bed load transport. Sometimes
sediments stay in suspension for an appreciable length of time called suspended load
transport. In addition, wash load is a bed material load according the particle size and
mainly moves as suspended load. So, sediment can be classified as bed material load
and wash load or bed load and suspended load. Wash load transport is a function of
basin characteristics, whereas bed load transport is a function of flow characteristics
(Yang 1996) (Figure 3.4).
33
Figure 3.4. Movement types of sediment particles
3.7.1. Bed Load Transport Formulas
Bed load motion starts when critical conditions are exceed. The motion
concerned with two phase (solid + liquid) flow near the bed. Generally, the bed load
transport rate of a river is about 5-25% of that in suspension. Bed load measurement is
difficult, so it is estimated by sediment transport formulas based on different modes of
motion employing different parameters, including shear stress and flow velocity. The
approaches for prediction of bed load are briefly summarized as follows.
3.7.1.1. DuBoys Approach
Duboys (1879) developed a bed load model using shear stress approach. This
model consists of sediment particles moving in layers because of the tractive force
acting along at the bed. The bed load capacity formula is given as;
( )cb Kq τττ −= (3.43)
where; Straub (1935) defined K coefficient depending on the sediment particle
characteristics.
34
4/3
173.0d
K = (3.44)
Thus DuBoys equation can be rewritten as,
( )cs
b dq τττ −= 4/3
173.0 (3.45)
where,
sd =sediment particle diameter in mm
τ and cτ =bed and critical shear stress respectively in Ib/ft2
bq =bed load transport capacity in (ft3/sec)/ft
3.7.1.2. Meyer – Peter’s Approach
Meyer-Peter et al. (1934) developed the following bed load formula using the
energy slope approach in metric system;
174.0 3/23/2
−=ss
b
dSq
dq (3.46)
where,
bq =bed load [in (kg/s)/m]
q =water discharge [in (kg/s)/m]
S =Slope
sd =particle size (in m)
Meyer – Peter formula is valid only for coarse material sediment particle
diameters greater then 3 mm. For mixtures of non uniform material, d should be
replaced by 35d , where 35% of the mixture is finer than 35d (Yang 1996).
35
3.7.1.3. Schoklitsch Formula
There are two Schoklitsch bed load formulas which were developed from
discharge approach. The first was published in 1934 in metric units.
)(7000 2/1
2/3
cs
b qqdSq −= (3.47)
where,
=bq bed load [in (kg/s)/m]
q and cq =water discharge and critical discharge at incipient motion [in m3/s)/m]
respectively
For sand with specific gravity 2.65, critical water discharge can be calculated by
plotting for given flow and grain diameter curve of bed load as ordinate against slope as
abscissa and then extrapolating the curve to zero bed load to obtain the intercept with
abscissa.
3/4
00001944.0S
dq sc = (3.48)
where,
sd =particle size (in m)
S =energy slope
The second bed load formula was published in 1943 in metric units.
)(2500 2/3cb qqSq −= (3.49)
For sand with specific gravity 2.65 critical water discharge can be calculated by
6/7
2/36.0S
dq sc = (3.50)
36
3.7.1.4. Shields Approach
Shields (1936) conducted laboratory studies and obtained the flow conditions
corresponding to incipient motion when sediment transport greater than zero. Shield’s
measurements provided semi empirical equation for estimating bed load transport
capacity (with English units);
( ) sws
c
w
sb
dSqq
γγττ
γγ
−−
= 10 (3.51)
where,
bq and q =bed load and water discharge per unit channel width, respectively
DSγτ =
cτ can be obtained from Shields diagram (Yang 1996).
3.7.1.5. Meyer – Peter and Müller’s Approach
Meyer-Peter and Müller (1948) transformed the Meyer-Peter bed load formula
by isolating involved parameters one by one.
3/23/12/3 25.0)(047.0)( bwsr
s qdRSKK ργγγ +−= (3.52)
where,
R =hydraulic radius (in m)
S =energy slope
d =mean particle diameter (in m)
=ρ specific mass of water (in metric ton – s/m4)
=bq bed-load rate in underwater weight per unit time and width [in (metric ton / s)/m]
=SKK rs )/( the kind of slope which is adjusted for energy loss due to grain resistance
rS , is responsible for the bed-load motion.
Energy slope can be calculated by Strickler (1923) formula:
37
3/42
2
RKVSs
= (3.53)
Energy loss due to grain resistance can be calculated by Strickler’s formula as:
3/42
2
RKVSr
r = (3.54)
So;
2/1
⎟⎠⎞
⎜⎝⎛=
SS
KK r
r
s (3.55)
The formula is based on a large quantity of experimental data. Test results
showed the relationship to be of the form;
SS
KK r
r
s =⎟⎟⎠
⎞⎜⎜⎝
⎛2/3
(3.56)
And
6/190
26d
Kr = (3.57)
where,
=90d the size of sediment for which 90% of the material is finer (Yang 1996).
3.7.1.6. Regression Approach
Rottner (1959) expressed bed load discharge in terms of the flow parameters
based on regression analysis. The formula is dimensionally homogeneous (Yang 1996).
38
( )[ ]
( )[ ]
33/250
3/250
2/1
2/13
778.014.0667.01
1
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎠⎞
⎜⎝⎛−
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛
−×
−=
Dd
Dd
gDV
gDq
s
ssb
ζ
ζγ
(3.58)
where,
bq =bed load discharge (in Ib/s per ft of width)
sγ =specific weight of sediment (in Ib/ft3)
sξ =specific gravity of the sediment (=2.65)
g =acceleration of gravity (in ft/s2)
D =mean depth (in ft)
V =mean velocity (in ft/s)
50d =particle size at which 50% of the bed material by weight is finer (in ft)
3.7.1.7. Chang, Simons and Richardson’s Approach
Chang, Simons and Richardson (1965) suggested that the bed load discharge by
weight can be determined using shear stress approach;
( )( ) φγγ
ττγtan−
−=
s
csbb
VKq (3.59)
( )ctb VKq ττ −= (3.60)
where,
bK =constant
tK =obtained by graph in English unit
φ =angle of repose of submerged material
bq =bed load transport capacity in Ib/ft/s (Yang 1996).
39
3.7.1.8. Parker et al. (1982) Approach
Parker et al. (1982) developed bed load transport formula using the hypothesis of
equal mobility.
ssi
bisi DgDp
qW 2/1
*
)()1/( −
=γγ
(3.61)
*
*
rττ
φ = (3.62)
*
*
ri
ii τ
τφ = (3.63)
*50
*50
50rτ
τφ = (3.64)
where,
=sD grain size
γ and =sγ specific weights of water and sediment
=biq bed-load per unit channel with in size fraction id
=ip friction by weight in size id
g=gravitational acceleration
=τ bed stress
sgDℜ= ρττ /* : Shields stress for grain size Ds
ii gDℜ= ρττ /* : Shields stress for i th grain size range
50*
50 / gDℜ= ρττ : Shields stress for median diameter of subpavement
=*rτ reference value of *τ at which **
rWW =
=*riτ reference value of i
*τ at which **ri WW =
=*50rτ 0.0876 reference value of *
50τ
40
Submerged=ℜ specific gravity of sediment
ρ = density of water
*)1/( riis
si d
Dτγγ
φ−
= (3.65)
The value of *riτ based on 50d is 0.0875
iri dd /0875.0 50* =τ (3.66)
[ ]25050
* )1(28.9)1(2.14exp0025.0 −−−= φφW , 0.95< 50φ <1.65 (3.67)
or
5.4
50
* 822.012.1 ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
φW , 50φ >1.65 (3.68)
where,
=*iW dimensionless bed-load in i th grain size sub range
=*W dimensionless total bed-load
3.7.1.9. Tayfur and Singh’s Approach
Tayfur and Singh (2006) derived sediment flux equation from Langbein and
Leopold (1968) sediment flux concentration equation. Tayfur and Singh obtained
following equation from the kinematic wave theory approach.
⎥⎦
⎤⎢⎣
⎡−=
max
1z
zzpVq sbs (3.69)
41
where,
=bsq the sediment flux in movable bed layer (L2/T)
=sV the velocity of sediment particles as concentration approaches zero (L/T)
maxz =the maximum bed elevation
z = the mobile bed layer elevation
p = porosity of the bed layer
3.7.2. Suspended Load Transport Formulas
Settling velocities are balanced by upward component of turbulent velocity and
stays in suspension. While particles fall, some of them are carried away with high flow
velocity and then returning near the bed region. Others particles caught in an upward
moving eddy are lifted again. The higher the turbulence, the smaller the particle size
and the greater the portion of the particles is lifted up. Thus some sediment is kept in
suspension. Some basic suspended load formulas are summarized below.
3.7.2.1. The Rouse Equation
The downward transport rate is settling by gravity and the upward transport rate
is rising by diffusion must be balanced under steady equilibrium conditions. Settling by
gravity and the rising by diffusion components in opposite direction, respectively are
Cω and dydC
sε where;
ω =fall velocity of sediment particles
C =sediment concentration
dydC =concentration gradient
sε =momentum diffusion coefficient for sediment. It is function of y.
In the form of balance;
0=+dydCC sεω (3.70)
42
Assuming the mass transfer coefficient is equal to momentum transfer coefficient;
ms βεε = (3.71)
where,
mε =momentum transfer coefficient
β =a factor of proportionality
In turbulent flows, the turbulent shear stress is;
dydu
my ρετ = (3.72)
A distance y above the bed is
( ) ⎟⎠⎞
⎜⎝⎛ −=−=
hyyhSy 1τγτ (3.73)
Assuming logarithmic velocity distribution;
kyU
dydu *= (3.74)
where,
u =local velocity at a distance y above the bed
*U =shear velocity
k =Prandtl-von Karman universal constant
From the equations below, mε and sε can obtained:
⎟⎠⎞
⎜⎝⎛ −=
hyykUm 1*ε (3.75)
⎟⎠⎞
⎜⎝⎛ −=
hyykUs 1*βε (3.76)
43
To integration Equation 3.70 a to y by substituting sε ,
∫ ∫⎟⎠⎞
⎜⎝⎛ −
−=y
a
y
a
hyykU
dyCdC
1*β
ω (3.77)
For fine sediments β can be assumed 1,
( )∫ −−=
y
aa
dyyhykU
hCC
*
ln ω (3.78)
When α is equal to;
( )yhykU −=
*
ωα (3.79)
So;
The Rouse (1937) equation becomes
α
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
aha
yyh
CC
a
(3.80)
3.7.2.2. Lane and Kalinske’s Approach
Lane and Kalinske (1941) assumed 1=β and obtained sediment concentration
in English units as;
⎟⎟⎠
⎞⎜⎜⎝
⎛=
hUa
PqCq Lasw*
15exp
ω (3.81)
44
where,
LP can be obtained by function of 6/1Dn and
*Uω with the results shown in Yang (1996)
book Figure 5.6.
3.7.2.3. Einstein’s Approach
Einstein (1950) replaced *U with '*U and assumed 1=β and obtained α as;
*4.0 Uω
α = (3.82)
Suspended load transported rate is
dycuqh
a∫= (3.83)
For any unit system
∫ ⎟⎠⎞
⎜⎝⎛
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
h
aa dyyU
aha
yyhCq 2.30log75.5 '
*
α
(3.84)
where,
xdxks // 65==Δ
Einstein (1950) obtain x and δ
sk that shown in Yang (1996) book Figure 3.9.
After substituting the logarithm velocity distribution formula simplifying one obtains:
⎥⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛
Δ= 21
'*
2.30log303.26.11 IIhaCUq a (3.85)
45
Clearly, 1I and 2I functions of A and their values can be obtained by numerical
integration with the results shown in Yang (1996) book Figure 5.7. and Figure 5.8.
Einstein (1950) assumed that da 2= , d is the representative grain size of bed material .
The concentration at ay = is
B
bwBwa au
qiAC 5= (3.86)
where,
bwBW qi =bed load transport rate by weight of size BWi
BU =average bed load velocity which was assumed by Einstein to be proportional to '*U
5A =a correction factor (=1/11.6)
Einstein’s equations can be applied to compute the suspended load discharge.
3.7.3. Total Load Transport Formulas
Total sediment load includes both bed load and suspended load. The transported
total bed material also consists of bed material load and wash load. However methods
for calculating the bed material load and wash load are different. The wash load is
estimated by measurements but since the bed surface is changing with incoming flow
continuously, it is difficult to predict the wash load in rivers. When comparing the
measured and computed total bed load, wash load should be removed from
measurements. Below, some basic total load formulas introduced.
3.7.3.1. Einstein’s Approach
Einstein (1950) obtained the bed load transport rate for per unit channel width;
( )
2/1
3*
−
⎥⎦
⎤⎢⎣
⎡−
=gd
giqis
sbwbwBW ρρρρφ (3.87)
46
Suspended load transport rate for per unit channel is
( )21 IIPqiqi EbwBWswsw += (3.88)
The total bed load for the size fraction ti ,
swswbwBWtt qiqiqi +=
( )211 IIPqi EbwBW ++= (3.89)
3.7.3.2. Laursen’s Approach
Laursen (1958) developed relationship between flow condition and sediment
discharge. ASCE Task Committee (1971) denoted Laursen’s formula in dimensionally
homogeneous form;
∑ ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠⎞
⎜⎝⎛=
i ici
iit
UfDdpC
ωττγ *
6/7
1'01.0 (3.90)
where,
tC =total average sediment concentration by weight per unit volume
gDSU =*
ip =percentage of materials available in size fraction i
iω =fall velocity of particles of mean size id in water
ciτ =critical tractive force for sediment size id as given by the Shields diagram,
3/1
502
58' ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
DdVρ
τ (3.91)
The bed material load is
tt qCq = (3.92)
47
where,
q =flow discharge per unit width
tq =dry weight of sediment discharge per unit time and width.
Laursen’s formula is applicable for fine sediment (Yang 1996).
3.7.3.3. Bagnold’s Approach
Bagnold (1966) estimated sediment transport capacity in submerged weight for
bed load and suspended load respectively as bellows:
bbww
ws Veq ταγ
γγ=
−tan (3.93)
01.0Vu
qs
sww
ws τωγγγ
=− (3.94)
where,
sγ and γ =specific weights of sediment and water, respectively
bwq =bed load transport rate by weight per unit channel width
swq =suspended load discharge in dry weight per unit time and channel width
tanα =ratio of tangential to normal shear force (can be obtained in graph that showed a
function of ( )[ ]dws γγτ −/ and tanα with grain size d in Yang (1996) book Figure
6.5.b.)
τ =shear force acting along the bed
V =average flow velocity
be =efficiency coefficient (can be obtained in graph that showed a function of V and be
with grain size d in Yang (1996) book Figure 6.5.a.)
ω =fall velocity for suspended sediment
su =mean transport velocity of suspended load
Transport function is based on stream power concept. Bagnold’s formula
expresses the bed load transport capacity by using energy concept as a function of work
48
done by system for transporting sediment (Yang 1996). The total load is sum of the bed
load as given the equation;
⎟⎠⎞
⎜⎝⎛ +
−=+=
ωατ
γγγ VeVqqq b
ws
wswbwz
01.0tan
(3.95)
where,
zq =total load [ ]ftsIbin //(
3.7.3.4. Engelund and Hansen’s Approach
Engelund and Hansen (1972) carried out Bagnold’s stream power concept and
obtained the sediment transport formula with similarity principle (Yang 1996);
2/51.0' θφ =⋅f (3.96)
where,
2
2'
VgSD
f = (3.97)
2/1
3
−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −= gdq s
st γ
γγγφ (3.98)
( )ds γγτ
θ−
= (3.99)
where,
g =gravitational acceleration
S =energy slope
V =average flow velocity
tq =total sediment discharge by weight per unit weight
49
d =median particle diameter
τ =shear stress along the bed
Engelund and Hansen’s formula is applicable only to dune beds and median
particle diameter is greater than 0.15 mm.
3.7.3.5. Ackers and White’s Approach
Ackers and White (1973) transport capacity formula based on Bagnold’s stream
power concept and applied dimensional analysis for uniform sediment. The
dimensionless sediment transport function is
n
sgr V
U
d
XDG ⎟⎠⎞
⎜⎝⎛= *
γγ (3.100)
where,
X =rate of sediment transport in terms of mass flow per unit mass flow rate (unitless)
D =water depth
*U =shear velocity
d =sediment particle size
V =average flow velocity
Ackers and White (1973) expressed the mobility number for sediment is
( )
n
sngr
dD
VgdUF
−−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
12/1
*/log32
1αγ
γ (3.101)
where,
n =transition exponent, depending on sediment size
α =coefficient in rough turbulent equation (=10)
The sediment size by a dimensionless grain diameter is
50
( ) 3/1
2
1/⎥⎦
⎤⎢⎣
⎡ −=
v
gdd s
gr
γγ (3.102)
where,
v =kinematic viscosity
The generalized dimensionless sediment transport function is
m
grgr A
FCG ⎟
⎟⎠
⎞⎜⎜⎝
⎛−= 1 (3.103)
The values, n , A , m and C can be obtained from laboratory data.
If 60>grd (3.104)
025.05.117.00.0
====
CmAn
If 601 ≤< grd (3.105)
( )14.023.0
log56.000.12/1 +=
−=−gr
gr
dA
dn
34.166.9+=
grdm
( ) ( ) 53.3loglog86.2log 2 −−= grgr ddC
Ackers and White’s approach applicable to mmd 04.0> and 8.0<rF .
where,
rF =Froude number
51
CHAPTER 4
ONE DIMENSIONAL HYDRODYNAMIC MODEL
The hydrodynamic model is described by equations of motion in open channel
flows. The flow model is developed to solve governing equations based on conservation
of mass and momentum. The flow depth and velocity of flow are sufficient to define the
flow conditions at a channel cross section, so two governing equations can be solved for
a typical flow situation.
In this Chapter, the continuity and momentum equations are derived that are
usually referred to as the Saint Venant equations.
4.1. de Saint Venant Equations
The one dimensional modeling of unsteady flow in open channels is most often
performed by supplementing de Saint Venant equations that describe the propagation of
a wave. In unsteady modeling, two flow variables, such as the flow depth and velocity
or the flow depth and the rate of flow are calculated to define the flow conditions at a
channel cross section. Therefore, two governing equations must be used to analyze a
typical flow situation. The required equations are the continuity equation and the
momentum equation derived with many assumptions (Roberson, et al. 1997, Chaudry
1993):
● The streamlines do not have sharp curves, so that the pressure distribution is
hydrostatic.
●As the channel bottom slope is small, the measured lateral and vertical velocity
are approximately same, so the lateral velocity and acceleration component can
be neglected.
●No lateral, secondary circulation occurs. The flow velocity distribution is
uniform over any channel cross section.
●The channel is prismatic with the same cross section and slope thorough out
the distance.
52
●The head losses in unsteady flow can be simulated by using the steady – state
resistance laws, so Chezy and Maning equations can be used also in unsteady
flow model. Water has uniform density and flow is generally subcritical
(Chaudhry 1993).
4.1.1. Continuity Equation in Unsteady Flows
According to the law of conservation of mass, both the difference of the rate of
mass inflow through area 1dA at section 1 and the rate of mass outflow through area
2dA at section 2 and the lateral inflow or outflow though xΔ in the same dt time space,
must be equal to changing of volume.
Figure 4.1. Definition sketch for continuity equation
( )( ) 02
1
1211122 =−−−+=∂∂
∫ dxxxqVAVAAdtd
tM x
x
ρρρρ (4.1)
where,
M =mass
A =flow area
21 1q
Flow Control Volume
xΔ x ( )111 ,, QVx ( )222 ,, QVx
dxxVV∂∂
+ V
53
V =flow velocity
ρ =mass density of water
1q =volumetric rate of lateral inflow or outflow per unit length of the channel between
sections 1 and 2. (inflow 1q is positive, outflow 1q is negative)
If water is assumed incompressible, mass density can be taken constant.
Therefore Equation (4.1) becomes,
( )( ) 02
1
1211122 =−−−+∫ dxxxqVAVAAdtd x
x
(4.2)
By rearranging Equation (3.2) with average flow area and average flow velocity
in channel cross section and applying the Leibnitz’s rule *, the equation becomes,
xdtqdtxxVVx
xAAAVdtdt
thxB Δ−⎥⎦
⎤⎢⎣⎡ Δ
∂∂
+⎥⎦⎤
⎢⎣⎡ Δ
∂∂
++−∂∂
Δ 1 (4.3)
where,
h =flow depth
The ⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
xV
xA can be neglected because it is small and with other simplifiers it
becomes,
01 =−∂∂
+∂∂
+∂∂ q
xAV
xVA
thB (4.4)
or
Bq
xhV
xV
BA
th 1=
∂∂
+∂∂
+∂∂ (4.5)
or
54
Bq
xhV
xVh
th 1=
∂∂
+∂∂
+∂∂ (4.6)
For a wide rectangular section the conservation of mass equation for water on a
unit with can be written as:
Bq
xhu
th w1=
∂∂
+∂∂ (4.7)
where,
h =the flow depth (L)
u =the flow velocity (L/T)
wq1 =the lateral flow flux (L2/T)
x =independent variable representing the coordinate in the longitudinal direction (flow
direction) (L)
t =independent variable of time (T)
*Leibnitz’s rule
( )( )
( )
( )
( )
( ) ( )( ) ( )( )dtdfttfF
dtdfttfFdxtxF
tdxtxF
dtd tf
tf
tf
tf
11
22 ,,,,
2
1
2
1
−+∂∂
= ∫∫
4.1.2. Momentum Equation in Unsteady Flows
The second required equation is derived by considering how the forces on the
control volume affect the movement of water through the control volume. The
momentum equation states that the rate of change of momentum is equal to the resultant
force acting on the control volume∑ =dtmvdF )( . In Figure 4.2 there is an element
which has mass m and length xΔ . The rate of changing of total momentum for that
element for the uncompressible flows is,
( )( )∑ ∫ Δ−−−+= xxxqVVAVVAVAdxVdtdF
x
xx
2
1
121111222 ρρρρ (4.8)
55
where,
xV =the component of the velocity of lateral inflow in the x - direction
1q = is positive in lateral inflow and negative in lateral outflow.
The Leibnitz’s rule must be applied to Equation (4.8) and VAQ = ,
( )∑ ∫ ⎟⎠⎞
⎜⎝⎛ −−−+
∂∂
= dxxxqVVQVQdxtQF
x
xx
2
1
1211122 ρρρρ (4.9)
The mean value theorem can save the Equation (4.9) to differential form, so
dividing the terms by ( )12 xx −ρ ,
( )( )
112
qVx
QVtQ
xxF
x−∂
∂+
∂∂
=−
∑ρ
(4.10)
Figure 4.2. Definition sketch for momentum equation
For the typical hydraulic engineering applications the shear stress on the flow
surface due to the wind and the effects of the Coriolis acceleration can be neglected.
The forces acting on the control volume are the pressure forces, the gravity force in the
x - direction and the frictional force which are explained below.
On the upstream end, the pressure force is;
111 hgAF ρ= (4.11)
56
On the downstream end, the pressure force;
222 hgAF ρ= (4.12)
where,
1h and 2h =depth of the centroid of flow area 1A and 2A , respectively
The weight of the water in the control volume in the x - direction is;
∫=2
1
03
x
x
dxASgF ρ (4.13)
where,
0S =The channel bottom slope
The frictional force due to water and the channel sides and channel bottom is;
dxASgFx
xf∫=
2
1
4 ρ (4.14)
where,
fS =The friction slope
The resultant force acting on the control volume is;
∑ −+−= 4321 FFFFF (4.15)
By inserting terms in the Equation (4.15) and dividing by ( )12 xx −ρ ,
( )( ) ( )dxSSA
xxg
xxhAhAg
xxF x
xf∫∑ −
−+
−−
=−
2
1
01212
2211
12ρ (4.16)
The mean value theorem can save the Equation (4.16) to differential form,
57
( ) ( ) ( )fx SSgAhAx
gqVx
QVtQ
−+∂∂
−=−∂
∂+
∂∂
01 (4.17)
By rearranging gives,
( ) ( ) 10 qVSSgAhgAQVxt
Qxf +−=+
∂∂
+∂∂ (4.18)
The Equation 4.18 is the momentum equation of water flow. If the right – hand
side of this equation is zero, it means that mass is conserved along any closed contour in
the tx − plane unless it is zero, this term acts like a source or a sink depending on 1q
(Cunge, et al. 1980).
( ) ( ) ( ) hAhBhhAhA −⎥⎦⎤
⎢⎣⎡ Δ−Δ+=Δ 2
21 (4.19)
So the higher – order terms can be neglected. Assuming 0→Δh , we can obtain
( )( ) AhAh =∂∂ and that can be rearranged as:
( ) ( )xhgA
xhhA
hghgA
x ∂∂
=∂∂
∂∂
=∂∂ (4.20)
The Equation 4.18 becomes,
( ) ( )fx SSgAqVxhgA
xQV
tQ
−+=∂∂
+∂
∂+
∂∂
01 (4.21)
The acceleration can be an increase in velocity at one point over time (local
acceleration) or an increase in velocity over space (acceleration may occur as water
moves along the control volume). These concepts lead to the de Saint Venant Equations,
the momentum equation, which when written in its conservation form is (Chaudhry
1993).
58
( )fSSghg
Vx
gtV
−=⎟⎟⎠
⎞⎜⎜⎝
⎛+
∂∂
+∂∂
0
2
2 (4.22)
Rearranging of the terms gives;
( ) 00 =−−∂∂
+∂∂
+∂∂
fSSgxhg
xVV
tV (4.23)
For steady uniform flow, the friction slope is equal to channel bottom slope. The
equation for steady, gradually varied flow is obtained by including the variation of the
flow depth and velocity head by including the derivative with respect to distance x . The
unsteadiness or the local acceleration term is needed to make the equation valid for
unsteady nonuniform flow model.
0SS f = xh∂∂
− ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−g
Vx 2
2
tV
g ∂∂
−1 (4.24)
de Saint Venant equations are nonlinear equations for which numerical methods
are required to solve them, so they were not practically applied in their full
Steady, nonuniform
Quasi-Steady Dynamic Wave Aprox.
Steady, uniform
Kinematic Wave Aprox.
Steady, nonuniform
Diffusion Wave Aprox.
Unsteady, nonuniform
Full Dynamic Wave Equation
Local Acceleration Term
Convective Acceleration Term
Pressure Force Term
Gravity Force Term
Friction Force Term
59
hydrodynamic form until the 1950s, although they were derived in the early nineteenth
century. A number of simplifications were performed by different researchers, being
more appropriate in particular situations. Consideration of the implications of the
different simplifications can also lead to a better understanding of the full equations so
de Saint Venant equations were described by the propagation of a wave. In wave
approximations, the continuity equation is solved simultaneously with approximate
form of the momentum equation. Their differences are all in momentum equation
assumptions. The three types of simplifications for wave models studied in this research
are summarized below.
4.2. Kinematic Wave Approximation
The kinematic wave approximation represents the change of flow with distance
and time by neglecting the local and convective acceleration terms of the momentum
equation. The assumption is that the water surface is parallel to the channel bed
(uniform flow assumption) in the kinematic wave approximation. It means there is no
way to represent backwater effects. These assumptions reduce the momentum equation
to the following.
fSS =0 (4.25)
The remaining terms represent the resistance equation for steady, uniform flow
as described by Manning’s or Chezy’s equation but can be taken into consideration the
effects of unsteadiness by an increase or decrease in the flow depth.
The resistance equation can be written as:
)(AfQ = (4.26)
By applying the chain rule we can write,
tQ
QA
tA
∂∂
∂∂
=∂∂ (4.27)
60
0xxQA
tQ
tA
=∂∂
∂∂
=∂∂ (4.28)
Substituting this equation into Equation 4.6, assuming 01 =q ,
0=∂∂
+∂∂
xqa
tQ (4.29)
where,
dAdQa = Represents the velocity of flood wave (L2/T)
This equation represents the kinematic wave which’s combined with continuity
equation. While Q is dependent variable, x and t are independent variables in this first
– order partial differential equation. If a is constant, the equation becomes linear. The
general solution of this linear equation by D’Alembert is,
)( atxfQ −= (4.30)
While 0=t , it represents initial conditions. The function creates a curve that
describes the variation of discharge Q with distance x . Assuming that t changes such as,
1t , 2t and 3t at velocity a in the downstream direction, the discharge occurs 1Q , 2Q and
3Q drawing a curve. It can be said that this curve always appears as )(xf , so it shows a
flood hydrograph in kinematic wave as it travels in the positive x - direction at velocity
a , the shape of the hydrograph does not change and its peak does not attenuate
(assuming a is constant). The kinematic wave equation describes the propagation of a
flood wave along a river reach but doesn't account for any backwater effects. This
implies that water may only flow downstream. The solution may be analytical or
numerical (Chaudhry 1993).
4.3. Diffusion Wave Approximation
The diffusion wave approximation is a simplified form of the dynamic wave
approximation. In addition, it is a significant improvement over the kinematic wave
61
model. In the diffusion wave approach, the th ∂∂ term from de Saint Venant equation
allows the water – surface slope to differ from the bed slope. This pressure differential
term allows the diffusion model to describe the attenuation of the floodwave. It also
allows the specification of a boundary condition at the downstream extremity of the
routing reach to account for backwater effects. The simplified form of the momentum
equation includes the convective acceleration term representing the spatial change in the
flow depth as well as the source terms, but neglects the temporal derivative term as well
as the convective acceleration terms due to spatial change in the flow velocity
(Chaudhry 1993). The simplified form of the momentum equation is,
)( 0 fSSxh
−=∂∂ (4.31)
Combining the simplified momentum equation with the continuity equation
gives the single equation called the diffusion wave equation.
xQa
tQ
xQD
∂∂
+∂∂
=∂∂
2
2
(4.32)
where,
02BSQD =
dAdQa =
The first term of the right side in this equation is the same as the equation for
kinematic model. The first term of the left side in Equation 4.32 represents the diffusion
of a flood wave as it travels in the channel. The coefficients D and a determined from
the observed hydrographs. By using them the attenuation of a flood wave due to storage
and friction can be included in the analysis (Chaudhry 1993).
4.4. Dynamic Wave Approximation
The dynamic wave equations are most accurate and comprehensive solution for
one dimensional unsteady flow problems in open channels under the specific
assumptions. The dynamic wave equations can be applied to wide range of one
dimensional flow problems such as dam break flood wave routing, evaluating flow
62
conditions due to tidal fluctuations, and routing flows through irrigation and canal
systems. The full equations can be solved by several numerical methods for incremental
times t and incremental distances x along the water way. The specific terms in the
momentum equation are small in comparison to the bed slope. In dynamic wave
approximation, the continuity equation is solved simultaneously with approximate form
of the momentum equation. If we reorganize the momentum equation Equation 4.24 the
full dynamic wave approximation can be defined by,
)( 0 fSSgxhg
xuu
tu
−=∂∂
+∂∂
+∂∂ (4.33)
where,
u =the flow velocity (L/T)
h =the flow depth (L)
=fS friction slope
=0S bed slope
g =acceleration due to gravity )/( 2TL
t =independent variable of time (T)
x =independent variable representing the coordinate in the longitudinal direction (flow
direction) (L)
63
CHAPTER 5
ONE DIMENSIONAL SEDIMENT TRANSPORT MODEL
The bed of the channel may aggrade or degrade in natural streams if the balance
of the water discharge or sediment is destroyed by natural or manmade factors. Eroding
loose surface from the basin by water deteriorates the ecology and changes the river
morphology. The water level rises and brings ecological problems when sediments are
deposited in river basins. It is essential to predict effects of sediment transport for river
management. Current research on river sediment transport prediction is mainly based on
numerical modeling of sediment transport. One dimensional unsteady sediment transport
models studied in two categories in this research: equilibrium and nonequilibrium.
5.1. One Dimensional Numerical Model for Sediment Transport under
Unsteady and Equilibrium Conditions
Bed material transportation is mathematically divided into two independent
processes: erosion and deposition. When the erosion and the deposition rates are equal
then there is equilibrium. It means that there is no interchange of sediment particles
between suspended and bed load sediment transport (Tayfur and Singh 2007). The
equilibrium condition exists when the same number of a given type and size of particles
are deposited on the bed as are entrained from it. In the literature, most of the studies are
based on equilibrium approach although natural rivers are mostly in nonequilibrium
state. When flow and sediment discharges, channel geometry and sediment properties
do not change substantially a long period of time, assuming the equilibrium sediment
transport conditions is appropriate.
5.1.1. Kinematic Wave Model of Bed Profiles in Alluvial Channels
under Equilibrium Conditions
The kinematic wave model neglects the local acceleration, convective
acceleration and pressure terms in the momentum equation for dynamic wave model.
64
Tayfur and Singh (2006) represented transport movement in a wide rectangular alluvial
channels as a system involving two layers: water flow layer and movable bed layer, as
shown in Figure 5.1. The water flow layer may contain suspended sediment. The
movable bed layer consists of both water and sediment particles, so the movable bed
layer includes porosity. The basic one dimensional partial differential equations for
unsteady and equilibrium nonuniform transport can be expressed as (Tayfur and Singh
2006):
Figure 5.1 Definition sketch of two layer system
(Source: Tayfur and Singh 2006)
● Continuity equation for water:
wqtzp
xchu
tch
1)1()1(
=∂∂
+∂−∂
+∂−∂ (5.1)
where,
u = flow velocity (L/T)
h = flow depth (L)
z = mobile bed layer elevation (L)
c = volumetric sediment concentration in the water flow phase (in suspension) ( 33 / LL )
wq1 =volumetric rate of lateral inflow or outflow of water (L/T)
p =porosity of the bed layer ( 33 / LL )
z
h water flow layer
movable bed layer
fixed bed layer
65
t =independent variable of time (T)
x =independent variable representing the coordinate in the longitudinal direction (flow
direction) (L)
● Continuity equation for sediment:
( ) sbs qx
qtzp
xhuc
thc
11 =∂∂
+∂∂
−+∂∂
+∂∂ (5.2)
where,
bsq =the sediment flux in the movable bed layer )/( 2 TL
sq1 =volumetric rate of lateral inflow or outflow of sediment (L/T)
For simplicity, if there is no lateral inflow of water and sediment, the terms on
the right hand sides of Equation 5.1 and Equation 5.2 vanish (Tayfur and Singh 2006).
Equations 5.1 and 5.2 contain five unknowns: zcuh ,,, and bsq . It means that
there must be three additional equations. One equation is obtained from momentum
equation for kinematic wave which is given as follows:
● Momentum equation for water:
of SS = (5.3)
Friction slope is taken as equal to bed slope employing Chezy’s equation for the
friction slope, yields;
1−= βαhu (5.4)
where, 5.0
fzSC=α
5.1=β
The second equation is obtained from Velikanov (1954), relating suspended
sediment concentration to flow variables as;
hgvuc
f
3κ= (5.5)
66
where,
g =gravitational acceleration (L/T2)
fv = average fall velocity of sediments (L/T)
κ = coefficient of sediment transport capacity
By rearranging the equation with fgv
κδ = , the equation becomes,
13 −= huc δ (5.6)
The third equation is obtained from Langbein and Leopold (1968) who proposed
a sediment flux concentration relation as:
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=
max
1b
bbsst C
CCvq (5.7)
where,
stq = sediment transport rate (M/L/T)
sv = velocity of sediment particles as concentration approaches zero (L/T)
bC = areal sediment concentration (M/L2)
maxbC = maximum areal sediment concentration (M/L2)
The sediment transport rate stq is in (M/L/T) (Equation 5.7) and the sediment flux
bsq is in (L2/T) (Equation 5.2), so stq is related to bsq as:
bssst qq ρ= (5.8)
where,
sρ = mass density of sediment (M/L3)
Areal concentration can be related to bed level as (Tayfur and Singh 2006):
( ) sb zpC ρ−= 1 (5.9)
67
By substituting Equations 5.8 and 5.9 into the Equation 5.7;
( ) ⎥⎦
⎤⎢⎣
⎡−−=
max
11z
zzvpq sbs (5.10)
where,
maxz = maximum bed elevation (L)
By using the chain rule, derivative of bsq can be obtained as:
( )xz
zzvp
xq
sbs
∂∂
⎥⎦
⎤⎢⎣
⎡−−=
∂∂
max
211 (5.11)
Hence the unknowns zcuh ,,, and bsq can be obtained by the system of
Equations 5.1, 5.2, 5.4, 5.5 and 5.11. After algebraic manipulation, the equations can be
written in compact form as:
[ ] [ ] 05.15.15.11 45.05.03 =∂∂
+∂∂
−+∂∂
−tzp
xhhh
thh δααδα (5.12)
[ ] [ ] ( ) ( ) 0211125.1max
45.03 =∂∂
⎥⎦
⎤⎢⎣
⎡−−+
∂∂
−+∂∂
+∂∂
xz
zzvp
tzp
xhh
thh sδαδα (5.13)
(note that 5.1=β )
5.1.1.1. Numerical Solution of Kinematic Wave Equations
In this model, a finite difference scheme developed by Lax (1954) is used. This
scheme can capture shocks, since all the governing equations are solved simultaneously.
There is no need for iterations when gradients are large. The Lax scheme is an explicit
scheme and does not require large matrices, so it is easy for solving general empirical
equations for roughness and sediment discharge. With reference to the finite difference
68
grid as shown in Figure 5.2, the partial derivatives and other variables are approximated
as follows.
Figure 5.2. Finite – difference grid
( )t
ffftf j
ij
ij
i
Δ+−
=∂∂ −+
+11
1 5.0 (5.14)
xff
xf j
ij
i
Δ−
=∂∂ −+
211 (5.15)
where,
i =the node in space
j =the node in time
xΔ and tΔ = the distance and steps, respectively.
Based on the finite difference approximation of Equations 5.14 and 5.15, 5.12
and 5.13 may be written as follows for determining the values 1+jh and 1+jz .
[ ][ ] ( )
[ ] ( )[ ]ji
ji
ji
ij
ji
ji
ij
ijijji
ji
ji
zzzh
p
hhxt
hhh
hhh
111
5.03
115.03
45.0
111
5.05.11
25.115.15.1
)(5.0
−++
−+−++
−−−
−
−ΔΔ
−−
−+=
δα
δαδαα
(5.16)
tΔ
i+1 i-1 i x
xΔ t
j+1
j
69
[ ] ( )[ ]( ) ( )[ ] ( )j
ij
iij
sj
ij
ij
iij
ji
ji
ijji
ji
ji
zzxt
zz
vhhhph
hhxt
ph
zzz
11max
111
5.03
11
4
111
22
15.01
5.12
2)(5.0
−+−++
−+−++
−ΔΔ
⎥⎦
⎤⎢⎣
⎡−−−−
−−
−ΔΔ
−+=
δα
δα
(5.17)
The hydrodynamic part of the model is:
[ ][ ] ( )j
ij
iij
ijijji
ji
ji hh
xt
hhh
hhh 115.03
45.0
111
25.115.15.1
)(5.0 −+−++ −
ΔΔ
−−
−+=δα
δαα (5.16a)
111 )( −++ = βα j
ij
i hu (5.16b)
By using the presented algorithm, the unknown values of h and z at the new
time level 1+j (future time) are determined from every interior node ( i = 2,…….,N-1).
The values of the dependent variables h and z at the boundary nodes 1 and N+1 are
determined by using boundary conditions. Also, at the time level j =1, initial conditions
are already known.
Initial conditions can be specified as:
( ) ohxh =0, (5.18)
( ) ozxz =0, (5.19)
where,
oh and oz = the initial flow depth (L) and the bed level (L), respectively.
The upstream boundary conditions can be specified as inflow hydrograph and
inflow sedimentograph.
( ) )(,0 thth = (5.20)
70
( ) )(,0 tztz = (5.21)
The downstream boundary conditions can be specified as:
( ) 0),(=
∂∂
xtNh ( 1
111
+−
++ = j
Nj
N hh ) 0.0>t (5.22)
( ) 0),(=
∂∂
xtNz ( 1
111
+−
++ = j
NjN zz ) 0.0>t (5.23)
●Stability
The numerical scheme has to satisfy the stability conditions. For this reason, the
Courant – Friedrichs – Lewy (CFL) condition was used. Since the water waves travel at
a much higher velocity than the bed transients this condition is given by:
( )1≤
ΔΔ+
=x
tghuCn (5.24)
where,
nC = Courant number (it was taken 2.0=nC in this research)
Equations 5.16 and 5.17 are solved simultaneously for each time step.
5.1.1.2. Model Testing for Hypothetical Cases
The hypothetical cases were analyzed assuming inflow hydrograph and
concentration at the upstream of the channel, shown in Figures 5.3a and 5.3b.
The channel was assumed to have a 1000 m length and 20 m width with 0.0025
bed slope. Chezy roughness coefficient is 50=zC m0.5/s. The sediment was assumed to
have 2650=sρ kg/m3, 32.0=sd mm, 48.0=p and sediment transport capacity
coefficient 000075.0=κ (Ching and Cheng 1964). Langbein and Leopold (1968)
suggest 245max =C kg/m2 (note that sb zpC ρ)1( −= ). In Figure 4.3b, 14=bC kg/m2
corresponds the bed level 01.0=z m and 140=bC kg/m2 corresponds the bed level
10.0=z m.
71
0
50
100
150
200
250
0 30 60 90 120 150 180 210 240 Time (min)
Q (m3/s)
020406080
100120140160
0 30 60 90 120 150 180 210 240 Time (min)
C (kg/m2)
Figure 5.3. (a) Inflow hydrograph (b) Inflow concentration
5.1.1.2.1. Hypothetical Case I: Effect of Inflow Concentration
Figure 5.4a shows that when the inflow concentration increases at the upstream
end of the channel, bed level gradually increases. In the Figure 5.4b when the
equilibrium feeding of the sediment occurs at the upstream, the bed level continues to
increase along the channel length. During the recession limp of the inflow concentration
the bed level starts to decrease toward the 10% length of the channel while it increases,
toward the 90% length of the channel (Figure 5.4c). Figure 5.4d shows that the bed
level decreases to the original level at the upstream section but as time progresses it
increases toward the downstream section.
a
b
72
0
0,02
0,04
0,06
0,08
0,1
0 200 400 600 800 1000
Distance (m)
Bed
Lev
el z
(m) 20 min
40 min60 min
0
0,02
0,04
0,06
0,08
0,1
0,12
0 200 400 600 800 1000
Distance (m)
Bed
Lev
el z
(m) 80 min
100 min
0
0,02
0,04
0,06
0,08
0,1
0 200 400 600 800 1000
Distance (m)
Bed
Lev
el z
(m)
120 min140 min160 min
0
0,02
0,04
0,06
0,08
0,1
0 200 400 600 800 1000
Distance (m)
Bed
Lev
el z
(m)
180 min200 min220 min240 min
Figure 5.4. Transient bed profile at (a) rising period (b) equilibrium period (c) recession period
(d) post recession period of inflow hydrograph and concentration
c
b
a
d
73
5.1.1.2.2. Hypothetical Case II: Effect of Particle Velocity and Effect
of Particle Fall Velocity
The objective of this case was to compare the sediment particle velocity and
particle fall velocity formulations employed in the developed model. The fall velocity
must be obtained for calculating the particle velocity. For that reason, first of all we
wanted to see four particle velocity formulation’s (in literature) performances under the
Rouse (1938)’s fall velocity formulation. The fall velocity value is for most natural
sands of the shape factor of 0.7 and 2.0=sd mm is 024.0=fv m/s (Rouse 1938).
Under the same particle fall velocity, the developed model was tested for four different
particle velocity formulations (Bor, et al. 2008).
One of the formulations is Chien and Wan (1999) formulation. For
mmd s 1008.0 << and 1550/10 << sdh , Chien and Wan (1999) presented the
following relation:
2
3)4.1/(u
uuv c
s −= (5.25)
where,
cu = critical flow velocity at the incipient sediment motion (L/T).
cu can be expressed as a function of the particle fall velocity fv and the shear
velocity Reynolds number *R as (Yang 1996):
⎪⎩
⎪⎨
⎧+
−=
f
ff
c
v
vR
vu
05.2
66.006.0)log(
5.2*
70702.1
*
*
>
<<
RR (5.26)
The shear velocity Reynolds number *R (Yang 1996):
υsdu
R ** = (5.27)
74
where,
υ = kinematic viscosity of water (L2/T)
*u = shear velocity (L/T) and defined as (Yang 1996):
oghSu =* (5.28)
The second selected formulation is Bridge and Dominic (1984) formulation that
is derived though a theoretical consideration of the dynamics of bed load motion.
)( ** cs uuv −= δ (5.29)
where,
cu* = critical shear velocity (L/T). Bridge and Dominic (1984) expressed the average
value of δ between 8 and 12. In this study the employed is 10=δ . The critical shear
velocity defined as (Bridge and Dominic 1984):
δφ 2
*
)(tanfc
vu = (5.30)
where,
=φtan the dynamic friction coefficient (average value between 0.48 and 0.58 (Bridge
and Dominic 1984). In this study =φtan 0.53 was employed.
The third selected was a constant particle velocity is smvs /010.0= .
The fourth particle velocity equation is Kalinske’s equation. Kalinske (1947)
assumed that
( )cs uubv −= (5.31)
where,
uvs , = instantaneous velocities of sediment and fluid
cu = critical flow velocity at incipient motion
b = a constant close to unity
75
Secondly we wanted to see these four particle velocity formulation’s
performances under the Dietrich (1982) fall velocity formulation (Bor, et al. 2008).
)(
3*2110 RRRW += (5.32)
where,
*W = the dimensionless fall velocity of the particle
The fall velocity of particle is (Dietrich 1982):
31
* )(⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
ρυρρ gW
v sf (5.33)
4*
3*
2**1
)(log00056.0
)(log00575.0)(log0982.0)(log929.1767.3
D
DDDR
+
−−+−= (5.34)
)6.4(log)1)(5.0(3.0
]6.4tanh[log)1(85.0
)1(1log
*2
*3.2
2
−−−+
−−−⎥⎦⎤
⎢⎣⎡ −−=
DCSFCSF
DCSFCSFR (5.35)
⎥⎦⎤
⎢⎣⎡ −+
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−=
5.2)5.3(1
*3 ]6.4tanh[log83.2
65.0
P
DCSFR (5.36)
where,
*D = the dimensionless particle size
CSF = the Corey shape factor. The mean value of CSF for most naturally occurring
sediment is between 0.5 and 0.8 (Dietrich 1982). This study employed a value of
65.0=CSF .
P = the Powers value of roughness (the average value of 6~5.3=P (Dietrich 1982)).
This study employed the value of 75.4=P .
The dimensionless particle size is expressed as (Dietrich 1982):
76
2
3
*)(
ρυρρ ss gd
D−
= (5.37)
The third selected particle fall velocity formulation is Yang (1996) formulation.
We wanted to see these four particle velocity formulation’s performances under the
Yang (1996)’s fall velocity formulation (Bor. et al, 2008).
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎥⎦
⎤⎢⎣
⎡ −
−
=
s
w
wss
s
w
ws
f
d
gdF
gd
v
32.3
)(
)(181
5.0
2
γγγ
υγγγ
mmd
mmdmm
mmd
s
s
s
0.2
0.21.0
1.0
>
≤<
≤
(5.38)
where,
5.0
3
25.0
3
2
)(36
)(36
32
⎪⎩
⎪⎨⎧
⎥⎦
⎤⎢⎣
⎡−
−⎥⎦
⎤⎢⎣
⎡−
+=wss
w
wss
w
gdgdF
γγγυ
γγγυ
(5.39)
In Figures 5.5a, 5.5b and 5.5c it is seen that while Kalinske (1947) and Bridge
and Dominic (1984) formula give a faster wavefront, constant smvs /01.0= and Chien
and Wan (1999) formula give slower wavefront in rising and equilibrium period. At
recession period, as the sediment feeding decreases the bed elevation starts to decrease
toward the upstream section (in 20% of the channel length) under constant
smvs /01.0= . It is seen that Kalinske (1947) and Bridge and Dominic (1984) formula
give similar performance and sediment moves faster towards downstream end. This is
reasonable, since the transient bed profile moves downstream and thus concentration
also increases downstream (Figure 5.5c). In the postrecession period, the bed level
increases to original bed level at the upstream section. It is seen that bed profile reached
original bed early with Kalinske (1947) and Bridge and Dominic (1984) formula
(Figure 5.5d) (Bor, et al. 2008).
The same simulations were obtained under the other three fall velocity
formulations.
77
t=80min the fall velocity is "Rouse"
0
0,02
0,04
0,06
0,08
0,1
0,12
0 100 200 300 400 500 600 700 800 900 1000Distance (m)
Bed
Lev
el z
(m)
Vs=0.010 m/sVs=Chien&WanVs=Bridge&DominicVs=Kalinske
t=40min the fall velocity is "Rouse"
0
0,01
0,02
0,03
0,04
0,05
0,06
0 100 200 300 400 500 600 700 800 900 1000Distance (m)
Bed
Lev
el z
(m)
Vs=0.010 m/sVs=Chien&WanVs=Bridge&DominicVs=Kalinske
t=160min the fall velocity is "Rouse"
0
0,02
0,04
0,06
0,08
0,1
0 100 200 300 400 500 600 700 800 900 1000Distance (m)
Bed
Lev
el z
(m)
Vs=0.010 m/sVs=Chien&WanVs=Bridge&DominicVs=Kalinske
t=240min the fall velocity is "Rouse"
0
0,02
0,04
0,06
0,08
0 100 200 300 400 500 600 700 800 900 1000Distance (m)
Bed
Lev
el z
(m)
Vs=0.010 m/sVs=Chien&WanVs=Bridge&DominicVs=Kalinske
Figure 5.5. Transient bed profile under different particle velocities at (a) rising period (b)
equilibrium period (c) recession period (d) postrecession period of inflow
hydrograph and concentration. (Source: under Rouse 1938, Dietrich 1982, Yang
1996 formula).
a
b
c
d
78
t=40 min "Chien & Wan" the particle velocity approach
0
0,01
0,02
0,03
0,04
0,05
0,06
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
Dietrich Yang Rouse
t=80 min "Chien & Wan" the particle velocity approach
0
0,02
0,04
0,06
0,08
0,1
0,12
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
Dietrich Yang Rouse
t=160 min "Chien & Wan" the particle velocity approach
0
0,02
0,04
0,06
0,08
0,1
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
Dietrich Yang Rouse
t=240 min "Chien & Wan" the particle velocity approach
0
0,02
0,04
0,06
0,08
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
Dietrich Yang Rouse
Figure 5.6. Transient bed profiles under different fall velocities at (a) rising period (b)
equilibrium period (c) recession period (d) postrecession period of inflow
hydrograph and concentration. (Source: under Chien and Wan 1999, Bridge and
Dominic 1984, Kalinske 1947 formulation.
a
b
c
d
79
In Figure 5.6a, 5.6b and 5.6c, the effect of the fall velocity on the sediment
transport under the different particle velocity formulations is given. Dietrich (1982),
Yang (1996) fall velocity formulations and constant smv f /024.0= value (Rouse
1938) give nearly same result under the Bridge and Dominic (1984), Kalinske (1947),
and Chien and Wan (1999) particle velocity formulation. For better assessment, the
model must be test with experimented results (Bor, et al. 2008).
The same simulation profiles were obtained under the other three particle
velocity formulations.
5.1.1.2.3. Hypothetical Case III: Effect of Maximum Concentration
In this case, different maximum concentration values were tested using
developed kinematic wave model. For that reason, 2max /840 mkgC = (corresponds to
maximum bed level of mz 60.0max = ), 2max /630 mkgC = (corresponds to maximum bed
level of mz 45.0max = ), 2max /420 mkgC = (corresponds to maximum bed level of
mz 30.0max = ) and 2max /245 mkgC = (corresponds to maximum bed level of
mz 17.0max = ) were selected respectively. It is seen in Figure 5.7 that higher maxz value
gives higher bed level in the transient bed form profile. The sediment particles move
faster downstream under high bed level. The bed levels increased gradually and
wavefronts moved slowly at the rising and equilibrium periods of the simulation (Figure
5.7a and 5.7b). At 160 min while the bed wavefront just reached about 400 m
under mz 15.0max = , it moved the downstream end under mz 60.0max = (Figure 5.7c).
While the front under mz 30.0max = closely followed the front under mz 17.0max = , front
under mz 45.0max = closely followed the front under mz 60.0max = (Figure 5.7a-5.7c).
80
0
0,01
0,02
0,03
0,04
0,05
0,06
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
t=40 min zmax=0.17 m t=40 min zmax=0.30 mt=40 min zmax=0.45 m t=40 min zmax=0.60 m
0
0,02
0,04
0,06
0,08
0,1
0,12
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
t=80 min zmax=0.17 m t=80 min zmax=0.30 mt=80 min zmax=0.45 m t=80 min zmax=0.60 m
0
0,02
0,04
0,06
0,08
0,1
0,12
0 200 400 600 800 1000
Distance (m)
Bed
Lev
el z
(m)
t=160 min zmax=0.17 m t=160 min zmax=0.30 mt=160 min zmax=0.45 m t=160 min zmax=0.60 m
0
0,02
0,04
0,06
0,08
0 200 400 600 800 1000Distance (m)
Bed
Lev
el z
(m)
t=240 min zmax=0.17 m t=240 min zmax=0.30 mt=240 min zmax=0.45 m t=240 min zmax=0.60 m
Figure 5.7. Transient bed profile under different maxz values at (a) rising period (b) equilibrium
period (c) recession period (d) postrecession period of inflow hydrograph and
concentration
d
c
a
b
81
5.1.2. Diffusion Wave Model of Bed Profiles in Alluvial Channels
under Equilibrium Conditions
The diffusion wave model neglects only the local and convective accelerations
in the dynamic wave momentum equation. It is the simplified form of the momentum
equation which includes also pressure force term. Thus, the momentum equation with
these simplifications for a wide rectangular alluvial channel with two layers (Figure 5.1)
becomes:
● Momentum equation for water
)( 0 fSSgxzg
xhg −=
∂∂
+∂∂ (5.40)
The flow velocity in open channels for diffusion waves can be calculated by
using either the Manning or Chezy’s formulations. We express as 1−= βαhu , α here
becomes:
5.0
⎟⎠⎞
⎜⎝⎛
∂∂
−∂∂
−=xz
xhSC ozα (5.41)
The algorithms for 1+jih and 1+j
iz is presented before by Equations 5.16 and 5.17.
The additional algorithm jfiS is determined as:
( ) ( )xzz
xhhSS
ji
ji
ji
ji
ojfi Δ
−−
Δ−
−= −+−+
221111 (5.42)
The hydrodynamic part of the model is:
( )xhhSS
ji
ji
ojfi Δ
−−= −+
211 (5.42a)
Equations 5.16, 5.17 and 5.42 are solved simultaneously for each time step.
82
5.1.2.1. Numerical Solution of Diffusion Wave Equation
Finite difference scheme developed by Lax (1954) is used in this model. The
partial derivatives were explained before in Equations 5.14 and 5.15. Initial and
boundary conditions were specified before Equations 5.18 – 5.23. And for stability the
Courant – Friedrichs – Lewy (CFL) condition was used.
5.1.3. Dynamic Model of Bed Profiles in Alluvial Channels under
Equilibrium Conditions
Conservation of mass equations for bed sediment in the movable bed layer,
considering there is no exchange of sediment due to the detachment and deposition
between two layers, and water for a wide rectangular alluvial channel:
● Continuity equation for water assuming that clear water ( )0=c :
wqxhu
xuh
th
1=∂∂
+∂∂
+∂∂ (5.43)
● Continuity equation for sediment:
sbs qx
qtzp
xhuc
thc
1)1( =∂∂
+∂∂
−∂∂
+∂∂ (5.44)
● Momentum equation for water
The one dimensional partial differential momentum equation of unsteady,
equilibrium flow in alluvial channel with dynamic wave assumption is;
)( 0 fSSgxzg
xhg
xuu
tu
−=∂∂
+∂∂
+∂∂
+∂∂ (5.45)
The friction slope fS in Equation 5.45 can be determined using the Chezy
equation (Equation 3.38).
83
5.1.3.1. Numerical Solution of Dynamic Wave Equations
In dynamic model, a finite difference scheme developed by Lax (1954) is used,
as explained as before. With reference to the finite difference grid as shown in Figure
5.2, additional to the partial derivatives, the variables are jih , j
iu , jiz and j
fiS are
approximated as follows:
( )ji
ji
ji hhh 112
1−+ += (5.46)
( )ji
ji
ji uuu 112
1−+ += (5.47)
( )jif
jif
jif SSS
1121
−++= (5.48)
Under the assumption there is no suspended sediment (clear water flow ( )0=c ),
the first and second term on the right side of the Equation 5.44 will disappear. Based on
the finite difference approximation of (5.14), (5.15), (5.46), (5.47) and (5.48), Equations
5.43 - 5.45 may be written as follows for determining the values 1+jh , 1+jiu and 1+jz :
( ) ( ) ( )ji
ji
ji
ji
ji
ji
ji
ji
ji hhu
xtuuh
xthhh 111111
1 5.05.05.0 −+−+−++ −
ΔΔ
−−ΔΔ
−+= (5.49)
( ) ( ) ( )
( ) ( )jfio
ji
ji
ji
ji
ji
ji
ji
ji
ji
ji
SStgzzgxt
hhgxtuuu
xtuuu
−Δ+−ΔΔ
−
−ΔΔ
−−ΔΔ
−+=
−+
−+−+−++
11
1111111
5.0
5.05.05.0 (5.50)
The hydrodynamic part of the model is:
( ) ( ) ( ) ( )jfio
ji
ji
ji
ji
ji
ji
ji
ji SStghhg
xtuuu
xtuuu −Δ+−
ΔΔ
−−ΔΔ
−+= −+−+−++
1111111 5.05.05.0 (5.50a)
84
( ) ( ) ( )ji
ji
ji
sj
ij
ij
i zzpz
zpv
xtzzz 11
max11
1
1
215.05.0 −+−+
+ −−
⎥⎦
⎤⎢⎣
⎡−
ΔΔ
−+= (5.51)
(note that bsq express as before Equation 5.11)
By using the presented algorithm, the unknown values of h and z at the new
time level 1+j (future time) are determined from every interior node ( i = 2,…….,N-1).
The values of the dependent variables h and z at the boundary nodes 1 and N+1 are
determined by using boundary conditions. Also, at the time level j =1, initial conditions
are already known.
Initial and boundary conditions were specified before Equations 5.18 – 5.23.
And for stability the Courant – Friedrichs – Lewy (CFL) condition was used.
Equations 5.49, 5.50 and 5.51 are solved simultaneously for each time step.
Note that, in the case of Dynamic Wave, we assumed that there is no suspended
sediment.
5.1.3.2. Model Testing: Comparing the Kinematic, Diffusion and
Dynamic Models for Hypothetical Cases
The hypothetical cases were analyzed assuming inflow hydrograph and
concentration at the upstream of the channel as shown in Figures 5.3a and 5.3b. The
channel was assumed to have a 1000 m length and 20 m width with 0.0025 bed slope.
Chezy roughness coefficient 50=zC m0.5/s and Manning roughness coefficient
31
021.0 −= smn . The sediment was assumed to have 2650=sρ kg/m3, 32.0=sd mm,
528.0=p and sediment transport capacity coefficient 000075.0=κ (Ching and Cheng,
1964). Langbein and Leopold (1968) suggest 245max =C kg/m2 (note that
sb zpC ρ)1( −= ).
For three of wave solutions a Courant number was selected 0.2. The numerical
solutions are plotted mx 200= , mx 500= and mx 800= along the channel, respectively
(Figure 5.8 and Figure 5.9). By comparing Figures 5.8a, 5.8b and 5.8c, one can observe
the different behavior of the diffusion and kinematic waves, particularly at peak flow
points. The diffusion wave reaches faster to maximum flow rate. On the other hand, the
85
dynamic wave has a smaller peak than the diffusion wave (Figure 5.9a, 5.9b and 5.9c)
and kinematic wave has the smallest. It can be said that particle velocity is higher in
diffusion and dynamic wave models. Results are acceptable with Kazezyılmaz et al.
(2007) paper.
x = 200 m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240
Time (min)
Bed
Ele
vatio
n z
(m)
Diffusion Wave Kinematic Wave
x = 500 m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240
Time (min)
Bed
Ele
vatio
n z
(m)
Diffusion Wave Kinematic Wave
x = 800 m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240Time (min)
Bed
Ele
vatio
n z
(m)
Diffusion Wave Kinematic Wave
Figure 5.8. Comparison of numerical solution of Diffusion and Kinematic waves at distance (a)
mx 200= (b) mx 500= (c) mx 800= of the channel
c
b
a
86
x = 200 m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240
Time (min)
Bed
Ele
vatio
n z
(m)
Kinematic Wave Diffusion Wave Dynamic Wave
x = 500 m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240Time (min)
Bed
Ele
vatio
n z
(m)
Kinematic Wave Diffusion Wave Dynamic Wave
x = 800 m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240
Time (min)
Bed
Ele
vatio
n z
(m)
Kinematic Wave Diffusion Wave Dynamic Wave
Figure 5.9. Comparison of numerical solution of Dynamic, Diffusion and Kinematic waves at
distance (a) mx 200= (b) mx 500= (c) mx 800= (assuming clear
water ( )0=c )
a
b
c
87
5.1.3.3. Hypothetical Case I: Comparing Three Bed Load Formulas
under Kinematic and Diffusion Wave Models
The objective of this case is to compare the bed load transport formulations
employed in the developed model. For that reason three bed load formulations were
selected from the literature. The formulations are Meyer – Peter (1934) (Equation 3.46),
Schoklitsch (1934) (Equation 3.47) and Tayfur and Singh (2006) (Equation 3.69) bed
load formulations. First of all, the formulations were tested under Kinematic wave
model. While Meyer – Peter and Schoklitsch formula give similar performance, Tayfur
and Singh formula gives different performance (Figure 5.10a and 5.10b). The sediment
particles moved downstream faster under Tayfur and Singh formula. The second test
was under Diffusion wave model, where the same behavior was observed (Figure 5.11a
and 5.11b).
200m
0
0,020,04
0,060,08
0,1
0 40 80 120 160 200 240Time (min)
Bed
Ele
vatio
n z
(m)
Tayfur&Singh Meyer-Peter Schoklitsch
160min
0
0,02
0,04
0,06
0,08
0,1
0 200 400 600 800 1000
Distance (m)
Bed
Ele
vatio
n z
(m)
Tayfur&Singh Meyer-Peter Schoklitsch
Figure 5.10. (a) Comparison of Tayfur and Singh, Meyer – Peter and Schoklitsch bed load
formulations under Kinematic wave model at time 160 min. (b) Comparison of
Tayfur and Singh, Meyer – Peter and Schoklitsch bed load formulations under
Kinematic wave model at distance mx 200= of the channel
a
b
88
200m
0
0,02
0,04
0,06
0,08
0,1
0 40 80 120 160 200 240Time (min)
Bed
Ele
vatio
n z
(m)
Tayfur&Singh Meyer-Peter Schoklitsch
160min
0
0,02
0,04
0,06
0,08
0,1
0 200 400 600 800 1000
Distance (m)B
ed E
leva
tion
z (m
)
Tayfur&Singh Meyer-Peter Schoklitsch
Figure 5.11. (a) Comparison of Tayfur and Singh, Meyer – Peter and Schoklitsch bed load
formulations under Diffusion wave model at time 160 min. (b) Comparison of
Tayfur and Singh, Meyer – Peter and Schoklitsch bed load formulations under
Diffusion wave model at distance mx 200= of the channel
It is seen that while Mayer – Peter (1934) and Schoklitsch (1934) formula give
same performance, Tayfur and Singh (2006) gives different. Sediment moves faster
towards under Tayfur and Singh (2006) formula.
5.1.3.4. Model Testing Using Experimental Data
5.1.3.4.1. Test I
The one dimensional model in tested by means of the experimental results
obtained by Bombar (Güney and Bombar 2008).These experiments are carried out on an
experimental system designed and constructed in the scope of TÜBİTAK project no:
106M274. The rectangular flume is 18.6 m long and 0.80 m wide. The bottom slope is
a
b
89
0.001. The input hydrograph constitute the upstream boundary condition. The
downstream boundary condition is defined by Equation 5.22 ( 11
11
+−
++ = j
Nj
N hh ).
The different input hydrographs in the form of isosceles triangle are generated
as shown in Figure 5.12. The steady discharge is 0.020 sm /3 while the peak discharge
value is equal to 0.060 sm /3 . The hydrographs with rising limb of 90 minutes and 120
minutes are given in Figure 5.12a and 5.12b respectively. The numerical equations,
Equation 5.16a and 5.16b are solved simultaneously for each time step under kinematic
wave approach. Equation 5.16a and 5.42a are solved simultaneously for each time step
under diffusion wave approach. Equation 5.49 and 5.50a are solved simultaneously for
each time step under dynamic wave approach.
Figures 5.13 and 5.14 represent the variations of water depths with time at
section 10.5 m and 15 m far from upstream end of the channel. These figures involve
the experiment results as well as those obtained from numerical solutions performed by
using various approaches; namely, kinematic diffusion and dynamic wave assumptions.
00,010,020,030,040,050,060,07
0 100 200 300 400 500Time (sec)
Q (m3/s)
00,010,020,030,040,050,060,07
0 100 200 300 400 500Time (sec)
Q (m3/s)
Figure 5.12. (a) The input hydrograph a) Rising limb = 90 second (b) Rising limb = 120 second
The results corresponding to the first hydrograph (rising limb = 90 sec) are given
in Figure 5.13 and those obtained from the second hydrograph (rising limb = 120 sec)
are depicted in Figure 5.14.
a
b
90
0,00
0,04
0,08
0,12
0,16
0,20
0 20 40 60 80 100 120 140 160 180 200Time (sec)
Flow
Dep
th h
(m)
Measured dataKinematic WaveDynamic WaveDiffusion Wave
0,00
0,04
0,08
0,12
0,16
0,20
0 20 40 60 80 100 120 140 160 180 200Time (sec)
Flow
Dep
th h
(m)
Measured dataKinematic WaveDynamic WaveDiffusion Wave
Figure 5.13. Measured and computed water depths at (a) 10.5 m (b) 14 m (the hydrograph that
has 90 second in rising limb)
0,00
0,04
0,08
0,12
0,16
0,20
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280Time (sec)
Flow
Dep
th h
(m)
Measured dataKinematic WaveDynamic WaveDiffusion Wave
0,00
0,04
0,08
0,12
0,16
0,20
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280Time (sec)
Flow
Dep
th h
(m)
Measured dataKinematic WaveDynamic WaveDiffusion Wave
Figure 5.14. Measured and computed water depths at (a) 10.5 m (b) 14 m (the hydrograph that
has 120 second in rising limb)
a
a
b
b
91
The overall computed error measures for simulations are presented in Table 5.1.
As seen, the mean relative error of 2.5=MRE implies that the developed model makes
about 5% error in predictions. The computed values of RMSE (root mean square error)
and MAE are 0.007 and 0.006 cm, respectively.
Table 5.1. Computed RMSE , MAE , MRE
Kinematic W. Dynamic W. Diffusion W.0,0074 0,0072 0,0073
Kinematic W. Dynamic W. Diffusion W.0,0065 0,0064 0,0064
Kinematic W. Dynamic W. Diffusion W.5,1952 5,0646 5,0789
RMSE, cm
MAE, cm
MRE, %
5.1.3.4.2. Test II
The second test was against the experimental data of aggradation depths
measured by Soni (1981a) in a laboratory flume of rectangular cross section. The flume
used by Soni was 30.0 m long, 0.20 m wide and 0.50 m deep. In the experimental run
constant equilibrium flow discharge was smQ /02.0 3= and uniform flow depth
was mh 092.00 = . The sand used for bed material and sediment feed in the experiments
had a median diameter of mmd s 32.0= and specific gravity of 2.65. Soni performed
experiments in the mobile bed condition to better represent natural rivers. Initially the
flume was filled with sand to a depth of 15 cm. Then the rectangular flume was filled
slowly with water and control valve was used to attain the specified discharge. The tail
gate height was adjusted in a way so that uniform flow was obtained in the flume by
allowing the bed to adjust by erosion or deposition. A uniform flow condition in the
flume was achieved when the measured bed and water surface were parallel to each
other. After reaching the uniform flow condition, sediment was dropped at the upstream
of the flume at a constant rate. The sediment injection section was located far enough
from the entrance of the flume to avoid entrance disturbances. The aggradation in the
bed started due to the excess load of the sediment. Bed and water surface elevation were
measured at regular time intervals (from 10 to 20 min at eleven sections). Aggradation
92
runs were continued until the end point of the transient profiles reached the downstream
end.
For computing the maximum bed elevation maxz , Langbein and Leopold (1968)’s
given a value of 2max /245 mkgCb = . The porosity was assumed to be 4.0=p . The flow
was uniform and steady and suspended sediment was negligible in this experiment, so
Equation 5.2 would suffice to model the bed aggradation.
Time : 30 min; Δqs=0.9qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24 26Distance (m)
Bed
leve
l (cm
)
Measured DataKinematic WaveEquilibrium bed profile
Time : 60 min; Δqs=0.9qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24 26Distance (m)
Bed
leve
l (cm
)
Measured DataKinematic WaveEquilibrium bed profile
Time : 90 min; Δqs=0.9qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24 26Distance (m)
Bed
leve
l (cm
)
Measured DataKinematic WaveEquilibrium bed profile
Figure 5.15. Simulation of measured bed profile at (a) 30 min (b) 60 min (c) 90 min
Figures 5.15a-5.15c show, respectively, simulations of bed profiles measured at
30, 60 and 90 min during the experimental run. The equilibrium flow conditions are
smQ /02.0 3= (equilibrium flow discharge), smqseq /10111 26−×= (equilibrium
a
b
c
93
sediment discharge), 00212.0=oS (bed slope), mho 092.0= (uniform flow depth) and
an excess sediment rate of seqs qq 9.0=Δ .
Figures 5.15 and 5.16 show the model simulations of the experiments. Figure
5.15 corresponds to the measured data under the rate of seqs qq 9.0=Δ . As seen that the
earlier parts of the transient profiles were closely captured by the model in downstream
end. It is observed that the transient profiles were faster than those of the measured ones
in reaching the equilibrium bed profile (Figure 5.16 and 5.16c). Figure 5.16
corresponds to the measured data under the rate of seqs qq 35.1=Δ . The simulations of
bed profiles measured at 15, 45, 75 and 105 min during the experimental run. The
measured and predicted profiles moved very closely toward the downstream end and
reached the equilibrium bed profile at the same time (Figure 5.16b). The measured and
predicted profiles moved together and reached the equilibrium bed profile at the same
time (Figure 5.16c). The predicted bed profile reached the equilibrium bed profile
slightly earlier than did the measured one (Figure 5.16d).
The overall computed error measures for simulations are presented in Table 5.2.
As seen, the mean relative error of 21.1=MRE implies that the developed model
makes about 1.21% error in predictions for seqs qq 9.0=Δ and for seqs qq 35.1=Δ the error
is 1.5%. The computed values of RMSE (root mean square error) and MAE are 0.89
and 0.75 cm, respectively. For seqs qq 35.1=Δ the computed values of RMSE and MAE
are 1.23 and 0.95 cm, respectively.
Table 5.2. Computed RMSE , MAE , MRE
RMSE, cm MAE, cm MRE, %Experiment KW KW KWΔqs=0.9qseq 0,89 0,75 1,21Δqs=1.35qseq 1,23 0,95 1,50
94
Time : 15 min; Δqs=1.35qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24Distance (m)
Bed
leve
l (cm
)Measured DataKinematic WaveEquilibrium bed profile
Time : 45 min; Δqs=1.35qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24Distance (m)
Bed
leve
l (cm
)
Measured DataKinematic WaveEquilibrium bed profile
Time : 75 min; Δqs=1.35qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24Distance (m)
Bed
leve
l (cm
)
Measured DataKinematic WaveEquilibrium bed profile
Time : 105 min; Δqs=1.35qs
545658606264666870
0 2 4 6 8 10 12 14 16 18 20 22 24Distance (m)
Bed
leve
l (cm
)
Measured DataKinematic WaveEquilibrium bed profile
Figure 5.16. Simulation of measured bed profile at (a) 15 min (b) 45 min (c) 75 min (d) 105
min
a
b
c
d
95
5.2. One Dimensional Numerical Model for Sediment Transport under
Unsteady and Nonequilibrium Conditions
All the sediment transport functions or equations presented earlier have been
intended for the estimation of bed levels at the equilibrium condition with no scour or
deposition, at least from a statistical point of view. It has been assumed that the amount
of wash load depends on the supply from upstream and is not a function of the hydraulic
conditions at a given station. Also, the amount of wash load is not high enough to
significantly affect the fall velocity of sediment particles, flow viscosity or flow
characteristics in a river in comparison with these values in clear water. When the wash
load or concentration of fine material is high, non equilibrium bed material sediment
transport may occur.
The floods may cause heavy erosion and landslides in a river basin causing
sediment overloading within a river reach. During the aggradation and degradation,
there may be an exchange of sediment particles between bed layer and suspended layer
exceeding the flow capacity. The nonequilibrium sediment transport condition results in
an unstable streambed elevation. In such cases a numerical sediment transport model
provides the computational framework for analysis.
There are significant differences between the calculations of equilibrium and
nonequilibrium conditions. The nonequilibrium condition solution can be obtained by
numerical sediment modeling using control volume approach.
5.2.1. Governing Equations
Tayfur and Singh (2007) studied transport movement in a wide rectangular
alluvial channels represented in two layers. Figure 5.17 shows the possible exchange of
sediment between two layers: the water flow layer and movable bed layer, depending
upon flow transport capacity and sediment rate in suspension.
96
Figure 5.17. Definition Sketch of two layer system in nonequilibrium condition
(Source: Tayfur and Singh 2007)
The water flow layer may contain suspended sediment. The movable bed layer
consists of both water and sediment particles; therefore bed layer includes porosity
(Tayfur and Singh 2007). Equations 5.1 and 5.2 are for equilibrium conditions, where
the entrainment rate ( zE ) is equal to the deposition rate ( cD ) ( cz DEei =.,. ). Under
nonequilibrium condition entrainment rate is not equal to the deposition rate ( cz DE ≠ ).
This makes the solution more complex than equilibrium approach. Pianese (1994)
employed one more equation, adaptation equation relating the change in bed level in
time to flow variables ( hu, ) , equilibrium suspended sediment concentration ( eqc ) and
suspended sediment concentration ( c ) to simplifing the solution. The adaptation
equation is,
( )ccuhtzp eq −=∂∂
−λ
)1( (5.52)
where,
λ =adaptation length
If the right hand side of the equation is negative, it represents detachment rate, if
it is positive, it represents deposition rate (Pianese 1994). When deposition occurs, z
increases, but c decreases. Otherwise, when detachment occurs, z decreases, c increases.
Mohammadian et al. (2004) employed an equation for the conservation of water
z
h Ez Dc
97
(Equation 5.1) (assuming clear water 0=c ) and an equation for conservation of
suspended sediment in the water flow layer as:
( )ccv
xchV
xxhuc
thc
eqf
x −+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
+∂∂
η (5.53)
where,
xV = the sediment mixing coefficient
η = a coefficient
As explained before, the right side of the equation represents deposition
(negative) or detachment rate (positive). They also used an additional equation which
represents the change in bed level in time to the particle fall velocity, equilibrium
suspended sediment concentration ( )eqc , and suspended sediment concentration ( )c as:
( ) ( )ccv
tzp eq
f −=∂∂
−η
1 (5.54)
There are some deficiencies in Equations 5.53 and 5.54. One of them is that
when the last term on the right hand side of the Equation 5.53 is negative, it acts a sink
of concentration in the bed layer, so there should be a negative sign in front of the term
on the right hand side of the equation. The second deficiency is concern with Equation
5.54. It does not fully represent the conservation of mass equation for the sediment in
the movable bed layer, since it ignores the major term of the sediment flux gradient
⎟⎠⎞
⎜⎝⎛∂∂
xqbs . Mohammadian et al. (2004) who did not employ Equation 5.2, ignored the bed
sediment flux term. To avoid any confusion, the conservation of mass for suspended
sediment in the water flow layer and the conservation of mass for bed sediment in the
movable bed layer are written separately (Tayfur and Singh 2007);
[ ]czs
sus DEqx
huct
hc−+=
∂∂
+∂∂
ρ1
1 (5.55)
98
( ) [ ]zcs
bedbs EDqx
qtzp −+=
∂∂
+∂∂
−ρ11 1 (5.56)
where,
susq1 = the lateral suspended sediment (L/T)
bedq1 =the lateral bed load sediment (L/T)
sρ = the sediment mass density (M/L3)
zE = the detachment rate (M/L2/T)
cD = the deposition rate (M/L2/T)
The equations include the exchange of sediment due to the detachment and
deposition between the two layers. The process is cz DE ≠ in the non equilibrium
condition. The process is cz DE = in the equilibrium condition. When cz DE > , there is
entrainment from the bed layer (reducing the bed elevation, increasing the suspended
sediment concentration). When cz DE < , there is deposition from the bed layer
(increasing the bed elevation, reducing the suspended sediment concentration).
According the Equations 5.1, 5.55 and 5.56, there are five unknowns, zcuh ,,,
and bsq . Therefore, two more equations are needed for solving the system. One more
equation can be obtained from the momentum equation for water flow. In this study, the
kinematic wave approximation was employed for the momentum equation (Equation
5.4). The fifth equation can be obtained by relating sediment transport rate to sediment
concentration in the movable bed layer. In this study, the kinematic theory was
employed (Tayfur and Singh 2006) (Equation 3.69).
Combining the Equations 3.69, 5.1, 5.4, 5.55 and 5.56 can be written in a
compact form as:
( ) ( ) ( ) ( )cq
xc
ch
tc
ch
tz
cp
xhh
th w
−=
∂∂
−−
∂∂
−−
∂∂
−+
∂∂
+∂∂ −
111111
ββ ααβ (5.57)
[ ]czs
sus DEhh
qxhhc
th
hc
xch
tc
−+=∂∂
−∂∂
+∂∂
+∂∂ −−
ραβα ββ 1121 (5.58)
99
( ) ( ) [ ]zcs
beds ED
ppq
xz
zzv
tz
−−
+−
=∂∂
⎥⎦
⎤⎢⎣
⎡−+
∂∂
11
121 1
max ρ (5.59)
These equations are kinematic wave equations for modeling unsteady state,
nonuniform transient channel bed profiles under nonequilibrium conditions. For
calculating the detachment rate zE , the shear stress approach was used (Yang 1996);
( )[ ]kcrcz TE ττσσ −Φ== (5.60)
where,
owhSγτ = (5.61)
( ) swscr dγγκτ −= (5.62)
where,
σ =the transfer rate coefficient (1/L)
cT = the flow transport capacity (M/L/T)
Φ = the soil erodibility coefficient
τ = the shear stress (M/L2)
crτ = the critical shear stress (M/L2)
k = an exponent
sw γγ , = the specific weight of water and sediment respectively (M/L3)
κ = a constant
sd = the sediment particle diameter (L)
The deposition rate cD can be expressed as (Yang 1996);
[ ]hucqD ssssc ρσσρ == (5.63)
where,
ssq = the unit suspended sediment discharge (M/L/T)
100
5.2.1.1. Numerical Solution of Kinematic Wave Equations
Equations 5.57, 5.58 and 5.59 were solved using the finite difference scheme
developed by Lax (1954) as explained before (Equations 5.14 and 5.15). Note that the
finite difference equations were written for both the layers not only at the central nodes
of the domain but also at the downstream nodes. All the equations were solved
simultaneously for each time step. The finite difference equations are:
( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ( )( ) ( )ji
wji
jij
i
ijji
ji
jij
i
ji
ji
ji
jij
i
ji
jiij
ji
ji
ji
ctqcc
ch
xtccc
ch
zzzc
phhhx
thhh
−Δ
+−−Δ
Δ++−
−+
+−−
−−Δ
Δ−+=
−+−++
−++
−+−
−++
1125.0
1
5.012
5.0
11111
1
111
111
111
β
β
α
αβ
(5.64)
( ) ( ) ( )[ ]
( ) [ ]ijc
ijzj
isj
i
susji
jiij
ji
ji
ji
jij
i
jij
ij
iijj
ij
ij
i
DEht
htqhhhc
xt
hhhhccch
xtccc
−Δ
+Δ
+−Δ
Δ−
+−−−ΔΔ
−+=
−+−
−++
−+−
−++
ραβ
α
β
β
111
2
111
111
111
2
5.02
5.0 (5.65)
( ) ( ) ( ) ( ) [ ]ijz
ijc
s
susji
ji
jisj
ij
ij
i EDp
tptqzz
zz
xtvzzz −
−Δ
+−Δ
+−⎟⎟⎠
⎞⎜⎜⎝
⎛−
ΔΔ
−+= −+−++
1121
25.0 1
11max
111
ρ (5.66)
where,
i = stands for space node
j = stands for time node
tΔ = time increment
xΔ =space increment
By using presented algorithm, the unknown values of ch, and z at the new time
level 1+j (future time) are determined at every interior node ( i = 2,….N-1). The values
of the dependent variables ch, and z at the boundary nodes 1 and N+1 are determined
by using boundary conditions. Also, at the time level j =1, initial conditions are already
known (Figure 5.2)
101
Initial conditions can be specified as:
( ) ohxh =0, (5.67)
( ) ocxc =0, (5.68)
( ) ozxz =0, (5.69)
where,
oo ch , and oz = the initial flow depth (L), concentration (L3/L3) and the bed level (L),
respectively.
The upstream boundary conditions can be specified as inflow hydrograph and
inflow sedimentograph.
( ) )(,0 thth = 0.0>t (5.70)
( ) )(,0 tctc = 0.0>t (5.71)
( ) )(,0 tztz = 0.0>t (5.72)
The downstream boundary conditions can be specified as:
( ) 0),(=
∂∂
xtNh ( 1
111
+−
++ = j
Nj
N hh ) 0.0>t (5.73)
( ) 0),(=
∂∂
xtNc ( 1
111
+−
++ = j
NjN cc ) 0.0>t (5.74)
( ) 0),(=
∂∂
xtNz ( 1
111
+−
++ = j
NjN zz ) 0.0>t (5.75)
102
●Stability
The numerical scheme has to satisfy the stability conditions. For this reason, the
Courant – Friedrichs – Lewy (CFL) condition was used. Since the water waves travel at
a much higher velocity than the bed transients this condition is given as before Equation
5.24.
Equations 5.64, 5.65 and 5.66 are solved simultaneously for each time step.
5.2.1.2. Model Application
The channel was assumed to have a 1000 m length and 30 m width with 0.0015
bed slope. The model parameters basically are as follows:
PCSFkzSpC ssoz ,,,,,,,,,,, max κφσγρ Φ and sd . Parameters sρ , sγ , and sd can be obtained
from experimental sediment data. Chezy roughness coefficient is assumed to be
smCz /36 5.0= . The sediment was assumed to have 3/2650 mkgs =ρ , mmds 32.0=
and 528.0=p . Maximum concentration was assumed 2max /500 mkgC = (note that
( ) spCz ρ−= 1maxmax ). Gessler (1965) suggested a value of 0.047 for κ for most flow
conditions. The value of transfer rate can be calculated in flumes by ( )h71=σ , where
h is flow depth, parameter Φ has a range of 0.0 – 1.0 and exponent ik has a range of 1.0
– 2.5 in literature (Foster 1982, Tayfur 2002, Yang 1996). The inflow hydrograph and
inflow concentration were given in Figure 5.18 for upstream boundary conditions.
103
0
20
40
60
80
100
120
140
0 30 60 90 120 150 180 210 240Time (min)
Q (m3/s)
0102030405060708090
0 30 60 90 120 150 180 210 240Time (min)
C (kg/m2)
Figure 5.18 (a) Inflow hydrograph. (b) Inflow concentration
Figures 5.19a-5.19d present bed profiles during the rising limp, equilibrium,
recession limb and postrecession limp of the inflow hydrograph and concentration,
respectively. It is seen that while inflow concentration increases, the bed level gradually
increases in upstream and it decreases after about 200 m in the downstream (Figure
5.19a). The bed elevation continues to increase in the equilibrium period at the upstream
end (Figure 5.19b). In rising period the bed level reaches at equilibrium after the 200 m
of the channel (Figure 5.21c). In procession period the bed level nearly same afert the
200 m (Figure 5.19d).
a
b
104
0
0,1
0,2
0,3
0,4
0,5
0 100 200 300 400 500 600 700 800 900 1000
Distance (m)
Bed
ele
vatio
n z
(m) 20 min
40 min60 min
0
0,1
0,2
0,3
0,4
0,5
0,6
0 100 200 300 400 500 600 700 800 900 1000
Distance (m)
Bed
ele
vatio
n z
(m)
80 min100 min
0
0,2
0,4
0,6
0,8
1
0 100 200 300 400 500 600 700 800 900 1000
Distance (m)
Bed
ele
vatio
n z
(m)
140 min160 min
0
0,2
0,4
0,6
0,8
1
0 200 400 600 800 1000
Distance (m)
Bed
ele
vatio
n z
(m)
180 min220 min240 min
Figure 5.19. Transient bed profile at (a) rising period (b) equilibrium period (c) recession
period (d) post recession period of inflow hydrograph and concentration
a
b
c
d
105
5.2.1.3. Model Testing Using Experimental Data
The model was tested against the experimental data of aggradation depths
measured by Yen et al. (1992) in laboratory flume. The flume used for present
experiments is 72 m long and 1 m wide. The water discharge was maintained at a
constant rate of sm /12.0 3 for all experiments. The initial bed slope is 0.0035 and
sediment median diameter is 1.8 mm. At the beginning of an experiment, a sediment
supply rate of 3.3 kg/min (dry mass) was continuously released from the upstream end
until the channel bed reached a state of equilibrium. The sediment supply rate was then
increased to 9.9 kg/min until a new equilibrium was reached. The rate of sediment
supply was thereafter reduced back to and kept at 3.3 kg/min until another new
equilibrium was reached. Finally, the sediment supply was cut off, and only clear water
was released from the upstream end until the channel bed was fully armored. Each
period lasted for about 30 hours. Bed elevations were measured 5 m apart from each
other. A sluice gate at the downstream end of the flume was employed to maintain a
constant tailwater level. The details of the experiment can be obtained from Yen et al.
(1992).
Simulations of bed profiles measured at 30, 60, 90 and 120 hours during the
experiment run (Figure 5.20.). The model and measured data nearly closed at each
location along the bed. At 120 hr predicted and measured data were nearly same (Figure
5.20d).
106
Time : 30 hr
2022242628303234
0 5 10 15 20 25Distance (m)
Bed
Ele
vatio
n z
(cm
) PredictedMeasuredOriginal Bed
Time : 60 hr
20
24
28
32
36
40
0 5 10 15 20 25Distance (m)
Bed
Ele
vatio
n z
(cm
) PredictedMeasuredOriginal Bed
Time : 90 hr
20222426
28303234
0 5 10 15 20 25Distance (m)
Bed
Ele
vatio
n z
(cm
) PredictedMeasuredOriginal Bed
Time : 120 hr
2022
24262830
3234
0 5 10 15 20 25Distance (m)
Bed
Ele
vatio
n z
(cm
) PredictedMeasuredOriginal Bed
Figure 5.20. Simulation of bed profiles along a channel bed at (a) 30 h, (b) 60 h, (c) 90
h and (d) 120 h of the laboratory experiment
a
b
c
d
107
Figure 5.21 presents simulation of bed level measured at 10 m away from the
upstream end during the experiment period of 120 hours. The model simulations of
transient bed levels at the specified locations are satisfactory. The model closely
predicted bed levels during, rising equilibrium and recession periods satisfactory.
The overall computed error measures for simulations are presented in Table 5.3
Location #1. As seen, the mean relative error of 99.1=MRE implies that the developed
model makes about 1.99% error in predictions. The computed values of RMSE (root
mean square error) and MAE are 0.82 and 0.66 cm, respectively.
Table 5.3. Computed RMSE , MAE , MRE
RMS, cm MAE, cm MRE, %0,820 0,663 1,992
Location #1
28
30
32
34
36
38
40
0 20 40 60 80 100 120
Time (hr)
Bed
Ele
vatio
n z
(cm
) PredictedMeasured
Figure 5.21. Simulation of bed profiles in time during the laboratory experiment at six
different locations of the experimental channel. Location #1 is 10 m away from
the upstream end (Yen, et al. 1992)
5.2.1.4. Model Testing: Comparing the Equilibrium and
Nonequilibrium models for Hypothetical Cases
The hypothetical cases were analyzed assuming inflow concentration
hydrograph at the upstream of the channel as shown in Figures 5.22. The channel was
assumed a flume and to have a 20 m length and 1 m width with 0.0001 bed slope. The
sediment was assumed to have 2650=sρ kg/m3, 09.0=sd mm, 45.0=p and sediment
transport capacity coefficient 000075.0=κ (Ching and Cheng 1964). Langbein and
108
Leopold (1968) suggest 500max =C kg/m2 (note that sb zpC ρ)1( −= ). The water
discharge is 5.0=Q sm /3 at the beginning. In equilibrium part smQ /1 3= (in
trapezoidal).
For two model solutions a Courant number was selected 0.2. The numerical
solutions are plotted mx 500= along the channel (Figure 5.23). It is seen that the
different behavior of equilibrium and nonequilibrium model, particularly at peak flow
points. The equilibrium model reaches faster to maximum flow rate. On the other hand,
the nonequilibrium model has a smaller peak. It can be said that bed material decreases
because of suspended sediment increases.
0,4
0,6
0,8
1
1,2
1,4
0 40 80 120 160 200 240
Time (min)
C (kg/m2)
Figure 5.22. (a) Inflow hydrograph. (b) Inflow concentration
Comparison Equilibrium and Nonequilibrium Model
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0,09
0,1
0 40 80 120 160 200 240
Time (min)
Bed
Lev
el z
(m)
nonequilibrium model equilibrium model
Figure 5.23. Comparing the equilibrium and nonequilibrium models
109
5.3. One Dimensional Numerical Model for Nonuniform Sediment
Transport under Unsteady and Nonequilibrium Conditions
One dimensional sediment transport models are simulated in uniform gravel bed
in this chapter. In this part, the proposed one dimensional model simulates the
nonequilibrium sediment transport of nonuniform total load under unsteady flow
conditions in rivers. For this reason, de Saint Venant equations are solved for complex
material. Models simulated suspended sediment transport using the nonequilibrium
transport approach. In this research, the mathematical model is developed using
diffusion wave theory under nonequilibrium condition. The bed profile evolution of
complex gravel in alluvial channels can be presented in Figure 5.24.
Figure 5.24. Multiply – layer model for bed load column
5.3.1. Governing Equations
The conservation of mass for suspended sediment in the water flow layer and the
conservation of mass for bed sediment in the movable bed layer separately can be
written for nonuniform and nonequilibrium sediment transport;
⎥⎦
⎤⎢⎣
⎡−+=
∂
∂+
∂
∂
∑∑∑∑
==
==N
kck
N
kzk
ssus
N
kk
N
kk
DEqx
chu
t
ch
111
11 1ρ
(5.76)
1sd
2sd
4sd
Z1
Z2
Z3
Z4
3sd
110
⎥⎦
⎤⎢⎣
⎡−+=
∂
∂+⎟
⎠⎞
⎜⎝⎛∂∂
⎟⎠
⎞⎜⎝
⎛− ∑∑
∑∑∑
==
=
==
N
kzk
N
kck
sbed
N
kbskN
k k
bN
kk EDq
x
q
tzp
111
1
11
11ρ
(5.77)
where,
kc = section – averaged sediment concentration of size class k
zkE = the detachment rate of size class k ( )TLM // 2
ckD = the deposition rate of size class k ( )TLM // 2
bskq = the sediment flux in the movable bed layer of size class k ( )TL /2
( )kb tz ∂∂ = bed change rate corresponding to the k th size class of sediment
kp = bed material porosity of size class k
5.3.1.1. Numerical Solutions of Nonuniform Model
Equations 5.40, 5.41, 5.76 and 5,77 were solved using the finite difference
scheme developed by Lax (1954) as explained before (Equations 5.14 and 5.15). The
finite difference equations are:
( )x
zz
xhhSS
N
kk
ji
N
kk
jij
ij
io
jfi Δ
⎟⎠
⎞⎜⎝
⎛−
−Δ−
−=∑∑=
−=
+−+
2
)()(
21
11
111 (5.78)
( ) ( )
⎟⎠
⎞⎜⎝
⎛−
Δ+⎟
⎠
⎞⎜⎝
⎛−
⎟⎠
⎞⎜⎝
⎛−
ΔΔ
+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛+−
⎟⎠
⎞⎜⎝
⎛−
+
⎥⎦
⎤⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛+−
⎟⎠
⎞⎜⎝
⎛−
−
−Δ
Δ−+=
∑∑∑
∑
∑∑∑∑
∑∑∑∑
=
=−
=+
=
=−
=+
=
+
=
=−
=+
=
+
=
−+−
−++
N
k
ji
wN
k
ji
N
k
jiN
k
ji
ij
N
k
ji
N
k
ji
N
k
jiN
k
ji
ji
N
k
ji
N
k
ji
N
k
jiN
k
ji
ji
jiij
ji
ji
ji
c
tqccc
hx
t
cccc
h
zzzc
p
hhhx
thhh
1
1
11
11
1
11
11
1
1
1
11
11
1
1
1
111
111
112
5.01
5.01
25.0
β
β
α
αβ
(5.79)
111
( ) ( )
( )[ ]
( )
[ ]kijck
ijzj
iksj
i
sus
ji
jiijk
ji
ji
ji
jij
i
kj
i
kj
ikj
iijkj
ikj
ikj
i
DEh
th
tq
hhhcx
t
hhhhc
cchx
tccc
)()()(
)(2
5.0)(
)()(2
)()(5.0
1
112
111
111
111
−Δ
+Δ
+
−Δ
Δ−
+−−
−ΔΔ
−+=
−+−
−++
−+−
−++
ρ
αβ
α
β
β
(5.80)
( ) ( )
( ) ( ) [ ]kijzk
ijc
kksk
sus
kj
ikj
ik
kj
iksk
jik
jik
ji
EDp
tp
tq
zzz
zx
vtzzz
)()(1)(1
)()()()(21
2)()()(5.0
1
11max
111
−−
Δ+
−Δ
+
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
ΔΔ
−+= −+−++
ρ
(5.81)
By using the presented algorithm, the unknown values of h and z at the new
time level 1+j (future time) are determined from every interior node ( i = 2,…….,N-1).
The values of the dependent variables h and z at the boundary nodes 1 and N+1 are
determined by using boundary conditions. Also, at the time level j =1, initial conditions
are already known.
Initial and boundary conditions were specified before Equations 5.18 – 5.23.
And for stability the Courant – Friedrichs – Lewy (CFL) condition was used.
Equations 5.49, 5.50 and 5.51 are solved simultaneously for each time step.
5.3.1.2. Model Application
The channel assumed as a flume has 20 m length and 1 m width with 0.0005 bed
slope. The model parameters basically are as follows:
PCSFkzSpC ssoz ,,,,,,,,,,, max κφσγρ Φ and sd . Parameters sρ , sγ , and sd can be obtained
from experimental sediment data. Chezy roughness coefficient is assumed to be
smCz /50 5.0= . It is assumed that there are four different sediment types in the sediment
column. Sediment characteristics that used in the model are summarized in Table 5.4.
112
Table 5.4. Sediment Characteristics
type ρs (kg/m3) ds (mm) p 1 2700 0.5 0.40 2 2650 0.7 0.45 3 2600 0.9 0.55 4 2500 1.2 0.60
Maximum concentration 2max /500 mkgC = was assumed for each particle sizes
(note that ( ) spCz ρ−= 1maxmax ). Gessler (1965) suggested a value of 0.047 for κ for
most flow conditions. The value of transfer rate can be calculated in flumes
by ( )h71=σ , where h is flow depth, parameter Φ has a range of 0.0 – 1.0 and
exponent ik has a range of 1.0 – 2.5 in literature (Foster 1982, Tayfur 2002, Yang
1996). The inflow hydrograph and inflow concentration were given in Figure 5.25 for
upstream boundary conditions.
1,3
1,5
1,7
1,9
2,1
2,3
2,5
0 50 100 150 200 Time (min)
Q (m3/s)
40
50
60
70
80
90
100
110
0 50 100 150 200 Time (min)
C (kg/m2)
Figure 5.25 (a) Inflow hydrograph. (b) Inflow concentration
a
b
113
40 min
0,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
100 min
0,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
160 min
0,10,2
0,30,40,5
0,60,70,8
0,91
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
240 min
0,10,20,3
0,40,50,60,7
0,80,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
Figure 5.26. Transient bed profiles of nonuniform sediment and uniform sediment model at (a)
rising period (b) equilibrium period (c) recession period (d) post recession period
of inflow hydrograph and concentration in unsteady flow conditions
a
b
c
d
114
Simulations were significantly under 50d (median diameter) and nonuniform
mixture for all the periods of the simulations. Under 50d (median diameter) conditions,
bed levels were lower than nonuniform flow case (Figure 5.26).
In another simulation for the same flume we considered constant inflow
hydrograph with smQ /2.1 3= and the same inflow sedimentograph seen in Figure
5.25.b. The simulations for the case are presented in Figure 5.27. While nonuniform
and uniform sediment transport model give similar performance under steady flow
condition, give different performance under unsteady flow conditions (Figure 5.27).
40 min
0,10,20,30,40,50,60,70,80,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
100 min
0,1
0,2
0,30,4
0,5
0,6
0,70,8
0,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
Figure 5.27. Transient bed profiles of nonuniform sediment and uniform sediment model at (a)
rising period (b) equilibrium period (c) recession period (d) post recession period
of inflow hydrograph and concentration in steady flow conditions
(cont. on next page)
a
b
115
160 min
0,1
0,20,3
0,4
0,50,6
0,7
0,80,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
240 min
0,1
0,20,3
0,4
0,50,6
0,7
0,80,9
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Distance (m)
Bed
Lev
el z
(m)
nonuniformmedian diameter
Figure 5.27. (cont.) Transient bed profiles of nonuniform sediment and uniform sediment
model at (a) rising period (b) equilibrium period (c) recession period (d) post
recession period of inflow hydrograph and concentration in steady flow
conditions
c
d
116
CHAPTER 6
SUMMARY AND CONCLUSION
6.1. Summary
Three mathematical and numerical models have been developed under
kinematic, diffusion and dynamic wave approaches for simulating bed profiles in
alluvial channels under unsteady and equilibrium conditions. Transient bed profiles are
also simulated for several hypothetical cases, comparing different particle velocities and
different particle fall velocities. The model tested with flume experiments. Also
different wave models (kinematic, diffusion and dynamic) were compared. The
kinematic wave model was developed for simulating transient bed profiles in alluvial
channels under unsteady and nonequilibrium conditions and tested against experimental
data. The diffusion wave model was developed for simulating transient bed profiles in
alluvial channels under unsteady, nonuniform and nonequilibrium conditions.
6.2. Conclusion
1. Numerical model is able to capture the effects of suspended sediment and bed
load sediment on the transport. When the transport capacity is greater than the
suspended load, deposition occurs, otherwise detachment occurs. The model is
able to capture this phenomenon.
2. The application of the developed model to hypothetical cases revealed that the
model is able to capture the behavior of the process in alluvial channels.
3. Modeling the process under nonequilibrium conditions give different results
than those under equilibrium conditions. Therefore, if the flow conditions in
nonequilibrium, it should be so modeled.
4. The model was not tested against experimental data under unsteady and
nonequilibrium flow and sediment loadings. The next aim is to test the model for
that general case.
117
5. The selected on particle velocity, particle fall velocity and hydrodynamic wave
(kinematic, diffusion and dynamic) would be better decided with testing of the
model with the general case (unsteady, nonequilibrium) experimental data.
6. Another shortcoming is the application of the model is field conditions. This is
able one of the future plans.
7. The investigation of different particle velocity formulations revealed that under
the same flow conditions, wave front is faster in Kalinske and Bridge and
Dominic’s formulation.
8. The investigation of different particle fall velocity formulations revealed that
under the same flow conditions, they produced nearly the same results.
9. The investigation of the effect of maxz (maximum bed level) on the transport
revealed that it is an important parameter. It significantly affects the wavefront
speed and bed level. The higher the maxz , the faster the wavefront and the higher
the bed level.
10. The numerical investigation of different sediment flux (bed load) formulations
revealed that under the same transport flow condition, the kinematic wave theory
produced different results then Meyer – Peter and Schoklists. Meyer – Peter and
Schoklists produced nearly the same profiles. Under kinematic wave theory, the
wavefronts move faster.
11. The numerical comparison of kinematic, diffusion and dynamic wave for
hypothetical cases of sediment transport revealed under the same sediment flux
function of the wavefront is slower in the case of kinematic wave.
12. The hydrodynamic part the developed numerical model was tested successfully
tested experimental flume data. It satisfactorily (less then 5%) simulated the
measured data.
13. The developed numerical model was tested against measured sediment data
from the literature. It predicted measured bed levels satisfactorily.
14. The numerical model revealed that modeling sediment mixtures with only mean
particle diameter 50d approximation might lead to misleading results. In other
words, it’s better model with the mixture with corresponding particle
characteristics i.e. 50d (median diameter), sρ (density) and p (porosity).
118
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APPENDIX A
CODES
Sub equilibrium() Dim h(502), u(502), z(502), hnew(502), znew(502) 'equilibrium' 'Kinematic wave approach' 'Vf Rouse, Vs Chien and Wan' 'DATA' L = 1000 'Channel Length' W = 20 'Channel Width' So = 0.0025 'Channel Slope' tn = 14400 dx = 2 dt = 0.1 nn = L / dx + 1 mm = tn / dt g = 9.81 K = 0.756 * 10 ^ -4 Cz = 50 Cmax = 245 p = 0.528 ros = 2650 ro = 1000 ds = 0.32 * 10 ^ (-3) al = Cz * So ^ 0.5 bet = 1.5 nu = 3 CSF = 0.65 pp = 4.75 fi = 0.53 trakd = 10 zmax = Cmax / (p * ros) vis = 1.139 * 10 ^ -6 'Initial and Boundary Conditions' 'Discharge Hydrograph' Q1 = 50 Q2 = 200 h1 = 1 h2 = 2.5198 'Sedimentgraph' c1 = 14 c2 = 140 'trapeziodal hydrograph' t1 = 0
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t2 = 600 t3 = 4200 t4 = 7200 t5 = 10800 t6 = 14400 ‘Time started t=0s’ t = 0 Do 'Initial Conditions' If t = 0 Then GoTo 1 Else GoTo 2 1 For i = 0 To nn z(i) = c1 / (p * ros) Next i For i = 0 To nn h(i) = h1 Next i GoTo 3 2 For i = 0 To nn z(i) = znew(i) Next i For i = 0 To nn h(i) = hnew(i) Next i 'Upstream Boundary Conditions' 3 If t < t2 Then GoTo 8 Else GoTo 9 8 h(0) = h1 z(0) = c1 / (p * ros) GoTo 17 9 If t2 <= t And t < t3 Then GoTo 10 Else GoTo 11 10 h(0) = 4.2217 * 10 ^ -4 * (t - 600) + h1 z(0) = (0.035 * (t - 600) + c1) / (p * ros) GoTo 17 11 If t3 <= t And t < t4 Then GoTo 12 Else GoTo 13 12 h(0) = h2 z(0) = c2 / (p * ros) GoTo 17 13 If t4 <= t And t < t5 Then GoTo 14 Else GoTo 15 14 h(0) = -4.2217 * 10 ^ -4 * (t - 10800) + h1 z(0) = (-0.035 * (t - 10800) + c1) / (p * ros) GoTo 17 15 h(0) = h1 z(0) = c1 / (p * ros)
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17 For i = 1 To nn u_ = (g * h(i) * So) ^ 0.5 Vf = 0.024 R_ = u_ * ds / vis If 1.2 < R_ < 70 Then uc_ = 2.5 * Vf / (Log(R_) - 0.06) + 0.66 * Vf If R_ > 70 Then uc_ = 2.05 * Vf u(i) = al * h(i) ^ (bet - 1) Vs = ((u(i) - (uc_ / 1.4) ^ 3 / u(i) ^ 2)) 'Chien & Wan' VON = K / (g * Vf) AA = 1 - VON * bet * al ^ 3 * h(i) ^ (bet - 1) BB = al * bet * h(i) ^ (bet - 1) - VON * bet * al ^ 4 * h(i) ^ (2 * bet - 2) CC = VON * bet * al ^ 3 * h(i) ^ (bet - 1) DD = VON * (2 * bet - 1) * al ^ 4 * h(i) ^ (2 * bet - 2) EE = p * Vs * (1 - 2 * z(i) / zmax) hnew(i) = 0.5 * (h(i + 1) + h(i - 1)) - dt * BB * (h(i + 1) - h(i - 1)) / (2 * dx * AA) znew(i) = 0.5 * (z(i + 1) + z(i - 1)) - dt * DD * (h(i + 1) - h(i - 1)) / (2 * p * dx) - dt * EE * (z(i + 1) - z(i - 1)) / (p * 2 * dx) - CC * (hnew(i) - 0.5 * (h(i + 1) + h(i - 1))) / p hnew(i) = 0.5 * (h(i + 1) + h(i - 1)) - dt * BB * (h(i + 1) - h(i - 1)) / (2 * dx * AA) - (1 - p) * (znew(i) - 0.5 * (z(i + 1) + z(i - 1))) / AA 'Downstream Boundary Conditions' hnew(nn) = hnew(nn - 1) znew(nn) = znew(nn - 1) dt = (dx / (u(i) + (g * hnew(i)) ^ 0.5)) * 0.1 Next i ‘New time’ t = t + dt Loop End Sub
Sub nonequilibrium() Dim h(502), u(502), z(502), hy(502), hk(502), zk(502), c(502), uk(502), hkapdate(502), ck(502), zg(502), Sf(502), uy(502), QQ(502), hky(502), unewp(502), unewpp(502), Ez(502), Dc(502) 'nonequilibrium sediment transport' 'kinematic Wave approach' 'DATA' 'channel length in m' L = 1000 'channel width in m' W = 30 'channel slope' So = 0.0015 tn = 14400
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dx = 2 dt = 0.1 nn = L / dx + 1 mm = tn / dt g = 9.81 K = 0.756 * 10 ^ -4 Cz = 36.5 Cmax = 500 p = 0.528 ros = 2650 ro = 1000 spww = ro * g spws = ros * g ds = 0.32 * 10 ^ (-3) 'm' bet = 1.5 nu = 3 CSF = 0.65 pp = 4.75 fi = 0.53 trakd = 10 vis = 1.139 * 10 ^ -6 n = 0.02 zmax = Cmax / (p * ros) 'Boundary and initial Conditions' 'Hydrographs' Q1 = 25 'dischare m^3/s' Q2 = 125 h1 = (Q1 / (W * Cz * (So) ^ 0.5)) ^ (1 / 1.5) h2 = (Q2 / (W * Cz * (So) ^ 0.5)) ^ (1 / 1.5) c1 = 80 'kg/m^2 sediment' c2 = 80 'trapeziodal hydrograph' 'in s' t1 = 0 t2 = 600 t3 = 4200 t4 = 7200 t5 = 10800 t6 = 14400 t = 0 'time started at t=0 s' Do 'initial conditions' If t = 0 Then GoTo 1 Else GoTo 2 1 For i = 0 To nn h(i) = h1 Next i For i = 0 To nn u(i) = Q1 / (h1 * W) Next i For i = 0 To nn z(i) = c1 / (p * ros)
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Next i For i = 0 To nn c(i) = c1 / ros Next i GoTo 3 2 For i = 0 To nn h(i) = hkapdate(i) Next i For i = 0 To nn z(i) = zk(i) Next i For i = 0 To nn c(i) = ck(i) Next i 'Boundary conditions upstream' 3 If t < t2 Then GoTo 8 Else GoTo 9 8 h(0) = h1 z(0) = c1 / (p * ros) u(0) = Q1 / (h1 * W) c(0) = c1 / ros GoTo 17 9 If t2 <= t And t < t3 Then GoTo 10 Else GoTo 11 10 h(0) = ((h2 - h1) * (t - t2)) / (t3 - t2) + h1 z(0) = (((c2 - c1) * (t - t2)) / (t3 - t2) + c1) / (p * ros) QQ(0) = ((Q2 - Q1) * (t - t2)) / (t3 - t2) + Q1 u(0) = QQ(0) / (h(0) * W) c(0) = (((c2 - c1) * (t - t2)) / (t3 - t2) + c1) / ros GoTo 17 11 If t3 <= t And t < t4 Then GoTo 12 Else GoTo 13 12 h(0) = h2 z(0) = c2 / (p * ros) QQ(0) = Q2 u(0) = Q2 / (h2 * W) c(0) = c2 / ros GoTo 17 13 If t4 <= t And t < t5 Then GoTo 14 Else GoTo 15 14 h(0) = ((h2 - h1) * (t - t5)) / (t4 - t5) + h1 z(0) = (((c2 - c1) * (t - t5)) / (t4 - t5) + c1) / (p * ros) QQ(0) = ((Q2 - Q1) * (t - t5)) / (t4 - t5) + Q1 u(0) = QQ(0) / (h(0) * W) c(0) = (((c2 - c1) * (t - t5)) / (t4 - t5) + c1) / ros GoTo 17 15 h(0) = h1 z(0) = c1 / (p * ros) u(0) = Q1 / (h1 * W) c(0) = c1 / ros
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17 For i = 1 To nn al = Cz * So ^ 0.5 bet = 1.5 Vf = 0.024 'fall velocity ROUSE' Vs = 0.01 'constant particle velocity' von = K / (g * Vf) 'velikanov' all = 0.5 'Flume' lamda = 7 * h(i) transferrate = 1 / lamda transferrate2 = all * Vf / (h(i) * u(i)) tto = 1000 * h(i) * So kk = 0.047 kısi = 0.5 ki = 2.5 ttocr = kk * (2650 - 1000) * ds Tc = (kısi * (tto - ttocr) ^ ki) 'detachment rate' Ez(i) = transferrate * Tc 'deposition rate' Dc(i) = transferrate2 * ros * h(i) * u(i) * c(i) 'Boundary conditions downstram' u(nn + 1) = u(nn - 1) h(nn + 1) = h(nn - 1) z(nn + 1) = z(nn - 1) c(nn + 1) = c(nn - 1) zk(i) = 0.5 * (z(i + 1) + z(i - 1)) - dt * Vs * (1 - 2 * z(i) / zmax) * (z(i + 1) - z(i - 1)) / (2 * dx) + dt * (Dc(i) - Ez(i)) / ((1 - p) * ros) hk(i) = 0.5 * (h(i + 1) + h(i - 1)) - dt * al * bet * h(i) ^ (bet - 1) * (h(i + 1) - h(i - 1)) / (2 * dx) - p * (zk(i) - 0.5 * (z(i + 1) + z(i - 1))) / (1 - c(i)) + dt * al * h(i) ^ bet * (c(i + 1) - c(i - 1)) / (2 * dx * (1 - c(i))) ck(i) = 0.5 * (c(i + 1) + c(i - 1)) - dt * al * bet * h(i) ^ (bet - 1) * (c(i + 1) - c(i - 1)) / (2 * dx) - c(i) * (hk(i) - 0.5 * (h(i + 1) + h(i - 1))) / h(i) - dt * al * bet * c(i) * h(i) ^ (bet - 2) * (h(i + 1) - h(i - 1)) / (2 * dx) + dt * (Ez(i) - Dc(i)) / (h(i) * ros) hkapdate(i) = 0.5 * (h(i + 1) + h(i - 1)) - dt * al * bet * h(i) ^ (bet - 1) * (h(i + 1) - h(i - 1)) / (2 * dx) - p * (zk(i) - 0.5 * (z(i + 1) + z(i - 1))) / (1 - c(i)) + dt * al * h(i) ^ bet * (c(i + 1) - c(i - 1)) / (2 * dx * (1 - c(i))) + h(i) * (ck(i) - 0.5 * (c(i + 1) + c(i - 1))) / (1 - c(i)) uk(i) = al * hkapdate(i) ^ 0.5 hkapdate(502) = hkapdate(500) ck(502) = ck(500) zk(502) = zk(500) 'stability' dt = (dx / (uk(i) + (g * hk(i)) ^ 0.5)) * 0.2